1. Introduction

The following notes aim to provide a very informal introduction to Stochastic Calculus, and especially to the Itô integral and some of its applications. They owe a great deal to Dan Crisan’s Stochastic Calculus and Applications lectures of 1998; and also much to various books especially those of L. C. G. Rogers and D. Williams, and Dellacherie and Meyer’s multi volume series ‘Probabilities et Potentiel’. They have also benefited from insights gained by attending lectures given by T. Kurtz.

The present notes grew out of a set of typed notes which I produced when revising for the Cambridge, Part III course; combining the printed notes and my own handwritten notes into a consistent text. I’ve subsequently expanded them inserting some extra proofs from a great variety of sources. The notes principally concentrate on the parts of the course which I found hard; thus there is often little or no comment on more standard matters; as a secondary goal they aim to present the results in a form which can be readily extended Due to their evolution, they have taken a very informal style; in some ways I hope this may make them easier to read.

The addition of coverage of discontinuous processes was motivated by my interest in the subject, and much insight gained from reading the excellent book of J. Jacod and A. N. Shiryaev.

The goal of the notes in their current form is to present a fairly clear approach to the Itô integral with respect to continuous semimartingales but without any attempt at maximal detail. The various alternative approaches to this subject which can be found in books tend to divide into those presenting the integral directed entirely at Brownian Motion, and those who wish to prove results in complete generality for a semimartingale. Here at all points clarity has hopefully been the main goal here, rather than completeness; although secretly the approach aims to be readily extended to the discontinuous theory. I make no apology for proofs which spell out every minute detail, since on a first look at the subject the purpose of some of the steps in a proof often seems elusive. I’d especially like to convince the reader that the Itô integral isn’t that much harder in concept than the Lebesgue Integral with which we are all familiar. The motivating principle is to try and explain every detail, no matter how trivial it may seem once the subject has been understood!

Passages enclosed in boxes are intended to be viewed as digressions from the main text; usually describing an alternative approach, or giving an informal description of what is going on – feel free to skip these sections if you find them unhelpful.

In revising these notes I have resisted the temptation to alter the original structure of the development of the Itô integral (although I have corrected unintentional mistakes), since I suspect the more concise proofs which I would favour today would not be helpful on a first approach to the subject.

These notes contain errors with probability one. I always welcome people telling me about the errors because then I can fix them! I can be readily contacted by email as alanb@chiark.greenend.org.uk. Also suggestions for improvements or other additions are welcome.

Alan Bain

2. Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . i
2. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
3. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 1
3.1. Probability Space . . . . . . . . . . . . . . . . . . . . . . . 1
3.2. Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 1
4. Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1. Stopping Times . . . . . . . . . . . . . . . . . . . . . . . 4
5. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.1. Local Martingales . . . . . . . . . . . . . . . . . . . . . . 8
5.2. Local Martingales which are not Martingales . . . . . . . . . . . 9
6. Total Variation and the Stieltjes Integral . . . . . . . . . . . . . 11
6.1. Why we need a Stochastic Integral . . . . . . . . . . . . . . . 11
6.2. Previsibility . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3. Lebesgue-Stieltjes Integral . . . . . . . . . . . . . . . . . . . 13
7. The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.1. Elementary Processes . . . . . . . . . . . . . . . . . . . . . 15
7.2. Strictly Simple and Simple Processes . . . . . . . . . . . . . . 15
8. The Stochastic Integral . . . . . . . . . . . . . . . . . . . . . 17
8.1. Integral for H ∈ L and M ∈ M_{2} . . . . . . . . . . . . . . . . 17
8.2. Quadratic Variation . . . . . . . . . . . . . . . . . . . . . 19
8.3. Covariation . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.4. Extension of the Integral to L^{2}(M) . . . . . . . . . . . . . . . 23
8.5. Localisation . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.6. Some Important Results . . . . . . . . . . . . . . . . . . . . 27
9. Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . 29
10. Relations to Sums . . . . . . . . . . . . . . . . . . . . . . . 31
10.1. The UCP topology . . . . . . . . . . . . . . . . . . . . . 31
10.2. Approximation via Riemann Sums . . . . . . . . . . . . . . . 32
11. Itô’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 35
11.1. Applications of Itô’s Formula . . . . . . . . . . . . . . . . . 40
11.2. Exponential Martingales . . . . . . . . . . . . . . . . . . . 41
12. Lévy Characterisation of Brownian Motion . . . . . . . . . . . 46
13. Time Change of Brownian Motion . . . . . . . . . . . . . . . 48
13.1. Gaussian Martingales . . . . . . . . . . . . . . . . . . . . 49
14. Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . 51
14.1. Change of measure . . . . . . . . . . . . . . . . . . . . . 51
15. Brownian Martingale Representation Theorem . . . . . . . . . 53
16. Stochastic Differential Equations . . . . . . . . . . . . . . . . 56
17. Relations to Second Order PDEs . . . . . . . . . . . . . . . . 61
17.1. Infinitesimal Generator . . . . . . . . . . . . . . . . . . . . 61
17.2. The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 62

17.3. The Cauchy Problem . . . . . . . . . . . . . . . . . . . . 64
17.4. Feynman-Kac̆ Representation . . . . . . . . . . . . . . . . . 66
18. Stochastic Filtering . . . . . . . . . . . . . . . . . . . . . . . 69
18.1. Signal Process . . . . . . . . . . . . . . . . . . . . . . . 69
18.2. Observation Process . . . . . . . . . . . . . . . . . . . . . 70
18.3. The Filtering Problem . . . . . . . . . . . . . . . . . . . . 70
18.4. Change of Measure . . . . . . . . . . . . . . . . . . . . . 70
18.5. The Unnormalised Conditional Distribution . . . . . . . . . . . 76
18.6. The Zakai Equation . . . . . . . . . . . . . . . . . . . . . 78
18.7. Kushner-Stratonowich Equation . . . . . . . . . . . . . . . . 86
19. Gronwall’s Inequality . . . . . . . . . . . . . . . . . . . . . . 87
20. Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 89
20.1. Conditional Mean . . . . . . . . . . . . . . . . . . . . . . 89
20.2. Conditional Covariance . . . . . . . . . . . . . . . . . . . . 90
21. Discontinuous Stochastic Calculus . . . . . . . . . . . . . . . 92
21.1. Compensators . . . . . . . . . . . . . . . . . . . . . . . . 92
21.2. RCLL processes revisited . . . . . . . . . . . . . . . . . . . 94
22. References . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3. Stochastic Processes

The following notes are a summary of important definitions and results from the theory of stochastic processes, proofs may be found in the usual books for example [Durrett, 1996].

3.1. Probability Space

Let (Ω, F, ℙ) be a probability space. The set of ℙ-null subsets of Ω is defined by

N := {N ⊂ Ω : N ⊂ A for A ∈ F, with ℙ(A) = 0} .

The space (Ω, F, ℙ) is said to be complete if for A ⊂ B ⊂ Ω with B ∈ F and ℙ(B) = 0 then this implies that A ∈ F.

In addition to the probability space (Ω, F, ℙ), let (E, E) be a measurable space, called the state space, which in many of the cases considered here will be (ℝ, B), or (ℝ^{n}, B). A random variable is a F/E measurable function X : Ω → E.

3.2. Stochastic Process

Given a probability space (Ω, F, ℙ) and a measurable state space (E, E), a stochastic process is a family (X_{t})_{t≥0} such that X_{t} is an E valued random variable for each time t ≥ 0. More formally, a map X : (ℝ^{+} × Ω, B^{+} ⊗ F) → (ℝ, B), where B^{+} are the Borel sets of the time space ℝ^{+}.

Definition 1. Measurable Process The process (X_{t})_{t≥0} is said to be measurable if the mapping (ℝ^{+} × Ω, B^{+} ⊗ F) → (ℝ, B) : (t, ω) ↦ X_{t}(ω) is measurable on ℝ × Ω with respect to the product σ-field B(ℝ) ⊗ F.

Associated with a process is a filtration, an increasing chain of σ-algebras i.e.

F_{s} ⊂ F_{t} if 0 ≤ s ≤ t < ∞.

Define F_{∞} by

F_{∞} = ⋁_{t≥0} F_{t} := σ (⋃_{t≥0} F_{t}) .

If (X_{t})_{t≥0} is a stochastic process, then the natural filtration of (X_{t})_{t≥0} is given by

F_{t}^{X} := σ(X_{s} : s ≤ t).

The process (X_{t})_{t≥0} is said to be (F_{t})_{t≥0} adapted, if X_{t} is F_{t} measurable for each t ≥ 0. The process (X_{t})_{t≥0} is obviously adapted with respect to the natural filtration.

Definition 2. Progressively Measurable Process A process is progressively measurable if for each t its restriction to the time interval [0, t], is measurable with respect to B_{[0,t]} ⊗ F_{t}, where B_{[0,t]} is the Borel σ algebra of subsets of [0, t].

Why on earth is this useful? Consider a non-continuous stochastic process X_{t}. From the definition of a stochastic process for each t that X_{t} ∈ F_{t}. Now define Y_{t} = sup_{s∈[0,t]} X_{s}. Is Y_{s} a stochastic process? The answer is not necessarily – sigma fields are only guaranteed closed under countable unions, and an event such as

{Y_{s} > 1} = ⋃_{0≤s≤s}{X_{s} > 1}

is an uncountable union. If X were progressively measurable then this would be sufficient to imply that Y_{s} is F_{s} measurable. If X has suitable continuity properties, we can restrict the unions which cause problems to be over some dense subset (say the rationals) and this solves the problem. Hence the next theorem. Theorem 3.3. Every adapted right (or left) continuous, adapted process is progressively measurable. Proof We consider the process X restricted to the time interval [0, s]. On this interval for each n ∈ ℕ we define

\begin{aligned}X_{1}^{n} & := ∑_{k=0}^{2^{n}−1} 1_{(ks/2^{n},(k+1)s/2^{n}]}(t)X_{ks/2^{n}}(ω), \\ X_{2}^{n} & := 1_{[0,s/2^{n})}(t)X_{0}(ω) + ∑_{k=1}^{2^{n}} 1_{[ks/2^{n},(k+1)s/2^{n})}(t)X_{(k+1)s/2^{n}}(ω) \end{aligned}

Note that X_{1}^{n} is a left continuous process, so if X is left continuous, working pointwise (that is, fix ω), the sequence X_{1}^{n} converges to X.

But the individual summands in the definition of X_{1}^{n} are by the adpatedness of X clearly B_{[0,s]} ⊗ F_{s} measurable, hence X_{1}^{n} is also. But the convergence implies X is also; hence X is progressively measurable.

Consideration of the sequence X_{2}^{n} yields the same result for right continuous, adapted processes.

The following extra information about filtrations should probably be skipped on a first reading, since they are likely to appear as excess baggage.

Define

\begin{matrix}∀t ∈ (0, ∞) F_{t−} = ⋁_{0≤s<t} F_{s} \\ ∀t ∈ [0, ∞) F_{t+} = ⋀_{t≤s<∞} F_{s}, \end{matrix}

whence it is clear that for each t, F_{t−} ⊂ F_{t} ⊂ F_{t+}. Definition 3.2. The family {F_{t}} is called right continuous if

∀t ∈ [0, ∞) F_{t} = F_{t+}.

Definition 3.3. A process (X_{t})_{t≥0} is said to be bounded if there exists a universal constant K such that for all ω and t ≥ 0, then |X_{t}(ω)| < K. Definition 3.4. Let X = (X_{t})_{t≥0} be a stochastic process defined on (Ω, F, ℙ), and let X′ = (X′_{t})_{t≥0} be a stochastic process defined on (Ω, F, ℙ). Then X and X′ have the same finite dimensional distributions if for all n, 0 ≤ t_{1} < t_{2} < ⋯ < t_{n} < ∞, and A_{1}, A_{2}, …, A_{n} ∈ E,

ℙ(X_{t_{1}} ∈ A_{1}, X_{t_{2}} ∈ A_{2}, …, X_{t_{n}} ∈ A_{n}) = ℙ′(X′_{t_{1}} ∈ A_{1}, X′_{t_{2}} ∈ A_{2}, …, X′_{t_{n}} ∈ A_{n}).

Definition 3.5. Let X and X′ be defined on (Ω, F, ℙ). Then X and X′ are modifications of each other if and only if

ℙ ({ω ∈ Ω : X_{t}(ω) = X′_{t}(ω)}) = 1 ∀t ≥ 0.

Definition 3.6. Let X and X′ be defined on (Ω, F, ℙ). Then X and X′ are indistinguishable if and only if

ℙ ({ω ∈ Ω : X_{t}(ω) = X′_{t}(ω)∀t ≥ 0}) = 1.

There is a chain of implications

indistinguishable ⇒ modifications ⇒ same f.d.d.

The following definition provides us with a special name for a process which is indistinguishable from the zero process. It will turn out to be important because many definitions can only be made up to evanescence. Definition 3.7. A process X is evanescent if ℙ(X_{t} = 0 ∀t) = 1.

4. Martingales

Definition 4.1. Let X = {X_{t}, F_{t}, t ≥ 0} be an integrable process then X is a

(i) Martingale if and only if 𝔼(X_{t}|F_{s}) = X_{s} a.s. for 0 ≤ s ≤ t < ∞ (ii) Supermartingale if and only if 𝔼(X_{t}|F_{s}) ≤ X_{s} a.s. for 0 ≤ s ≤ t < ∞ (iii) Submartingale if and only if 𝔼(X_{t}|F_{s}) ≥ X_{s} a.s. for 0 ≤ s ≤ t < ∞ Theorem (Kolmogorov) 4.2. Let X = {X_{t}, F_{t}, t ≥ 0} be an integrable process. Then define F_{t+} := ⋀_{ϵ>0} F_{t+ϵ} and also the partial augmentation of F by F̃_{t} = σ(F_{t+}, N). Then if t ↦ 𝔼(X_{t}) is continuous there exists an F̃_{t} adapted stochastic process X̃ = {X̃_{t}, F̃_{t}, t ≥ 0} with sample paths which are right continuous, with left limits (CADLAG) such that X and X̃ are modifications of each other. Definition 4.3. A martingale X = {X_{t}, F_{t}, t ≥ 0} is said to be an L^{2}-martingale or a square integrable martingale if 𝔼(X_{t}^{2}) < ∞ for every t ≥ 0. Definition 4.4. A process X = {X_{t}, F_{t}, t ≥ 0} is said to be L^{p} bounded if and only if sup_{t≥0} 𝔼(|X_{t}|^{p}) < ∞. The space of L^{2} bounded martingales is denoted by M_{2}, and the subspace of continuous L^{2} bounded martingales is denoted M_{2}^{c}. Definition 4.5. A process X = {X_{t}, F_{t}, t ≥ 0} is said to be uniformly integrable if and only if

sup_{t≥0} 𝔼 (|X_{t}|1_{|X_{t}|≥N}) → 0 as N → ∞.

Orthogonality of Martingale Increments A frequently used property of a martingale M is the orthogonality of increments property which states that for a square integrable martingale M, and Y ∈ F_{s} with 𝔼(Y^{2}) < ∞ then

𝔼 [Y (M_{t} − M_{s})] = 0 for t ≥ s.

Proof Via Cauchy Schwartz inequality 𝔼|Y (M_{t} − M_{s})| < ∞, and so

𝔼(Y (M_{t} − M_{s})) = 𝔼 (𝔼(Y (M_{t} − M_{s})|F_{s})) = 𝔼 (Y 𝔼(M_{t} − M_{s}|F_{s})) = 0.

A typical example is Y = M_{s}, whence 𝔼(M_{s}(M_{t} − M_{s})) = 0 is obtained. A common application is to the difference of two squares, let t ≥ s then

\begin{matrix}𝔼((M_{t} − M_{s})^{2}|F_{s}) = & 𝔼(M_{t}^{2}|F_{s}) − 2M_{s}𝔼(M_{t}|F_{s}) + M_{s}^{2} \\ = & 𝔼(M_{t}^{2} − M_{s}^{2}|F_{s}) = 𝔼(M_{t}^{2}|F_{s}) − M_{s}^{2}. \end{matrix}

4.1. Stopping Times

A random variable T : Ω → [0, ∞) is a stopping (optional) time if and only if {ω : T (ω) ≤ t} ∈ F_{t}.

The following theorem is included as a demonstration of checking for stopping times, and may be skipped if desired. Theorem 4.6. T is a stopping time with respect to F_{t+} if and only if for all t ∈ [0, ∞), the event {T < t} if F_{t} measurable. Proof If T is an F_{t+} stopping time then for all t ∈ (0, ∞) the event {T ≤ t} is F_{t+} measurable. Thus for 1/n < t we have

\{T ≤ t − \frac{1}{n}\} ∈ F_{(t−1/n)^{+}} ⊂ F_{t}

so

\{T < t\} = ⋃_{n=1}^{∞} \{T ≤ t − \frac{1}{n}\} ∈ F_{t}.

To prove the converse, note that if for each t ∈ [0, ∞) we have that {T < t} ∈ F_{t}, then for each such t

\{T < t + \frac{1}{n}\} ∈ F_{t+1/n},

as a consequence of which

\{T ≤ t\} = ⋂_{n=1}^{∞} \{T < t + \frac{1}{n}\} ∈ ⋀_{n=1}^{∞} F_{t+1/n} = F_{t+}.

Given a stochastic process X = (X_{t})_{t≥0}, a stopped process X^{T} may be defined by

\begin{matrix}X^{T} (ω) := & X_{T (ω)∧t}(ω), \\ F_{T} := & \{A ∈ F : A ∩ \{T ≤ t\} ∈ F_{t}\}. \end{matrix}

Theorem (Optional Stopping). Let X be a right continuous integrable, F_{t} adapted process. Then the following are equivalent:

(i) X is a martingale. (ii) X^{T} is a martingale for all stopping times T. (iii) 𝔼(X_{T}) = 𝔼(X_{0}) for all bounded stopping times T. (iv) 𝔼(X_{T} |F_{S}) = X_{S} for all bounded stopping times S and T such that S ≤ T. If in addition, X is uniformly integrable then (iv) holds for all stopping times (not necessarily bounded).

The condition which is most often forgotten is that in (iii) that the stopping time T be bounded. To see why it is necessary consider B_{t} a Brownian Motion starting from zero. Let T = inf{t ≥ 0 : X_{t} = 1}, clearly a stopping time. Equally B_{t} is a martingale with respect to the filtration generated by B itself, but it is also clear that 𝔼B_{T} = 1 ≠ 𝔼B_{0} = 0. Obviously in this case T < ∞ is false. Theorem (Doob’s Martingale Inequalities). Let M = {M_{t}, F_{t}, t ≥ 0} be a uniformly integrable martingale, and let M^{∗} := sup_{t≥0} |M_{t}|. Then

(i) Maximal Inequality. For λ > 0,

λℙ(M^{∗} ≥ λ) ≤ 𝔼 [|M_{∞}|1_{M^{∗}<∞}] .

(ii) L^{p} maximal inequality. For 1 < p < ∞,

∥M^{∗}∥_{p} ≤ \frac{p}{p − 1}∥M_{∞}∥_{p}.

Note that the norm used in stating the Doob L^{p} inequality is defined by

∥M∥_{p} = [𝔼(|M|^{p})]^{1/p} .

Theorem (Martingale Convergence). Let M = {M_{t}, F_{t}, t ≥ 0} be a martingale.

(i) If M is L^{p} bounded then M_{∞}(ω) := lim_{t→∞} M_{t}(ω) ℙ-a.s. (ii) If p = 1 and M is uniformly integrable then lim_{t→∞} M_{t}(ω) = M_{∞}(ω) in L^{1}. Then

for all A ∈ L^{1}(F_{∞}), there exists a martingale A_{t} such that lim_{t→∞} A_{t} = A, and A_{t} = 𝔼(A|F_{t}). Here F_{∞} := lim_{t→∞} F_{t}.

(iii) If p > 1 i.e. M is L^{p} bounded lim_{t→∞} M_{t} = M_{∞} in L^{p}. Definition 4.7. Let M_{2} denote the set of L^{2}-bounded CADLAG martingales i.e. martingales M such that

sup_{t≥0} 𝔼M_{t}^{2} < ∞.

Let M_{2}^{c} denote the set of L^{2}-bounded CADLAG martingales which are continuous. A norm may be defined on the space M_{2} by ∥M∥^{2} = ∥M_{∞}∥_{2}^{2} = 𝔼(M_{∞}^{2}).

From the conditional Jensen’s inequality, since f(x) = x^{2} is convex,

\begin{matrix}𝔼 (M_{∞}^{2}|F_{t}) ≥ & (𝔼(M_{∞}|F_{t}))^{2} \\ 𝔼 (M_{∞}^{2}|F_{t}) ≥ & (𝔼M_{t})^{2}. \end{matrix}

Hence taking expectations

𝔼M_{t}^{2} ≤ 𝔼M_{∞}^{2},

and since by martingale convergence in L^{2}, we get 𝔼(M_{t}^{2}) → 𝔼(M_{∞}^{2}), it is clear that

𝔼(M_{∞}^{2}) = sup_{t≥0} 𝔼(M_{t}^{2}).

Theorem 4.8. The space (M_{2}, ∥·∥) (up to equivalence classes defined by modifications) is a Hilbert space, with M_{2}^{c} a closed subspace. Proof We prove this by showing a one to one correspondence between M_{2} (the space of square integrable martingales) and L^{2}(F_{∞}). The bijection is obtained via

\begin{matrix}f : & M_{2} → L^{2}(F_{∞}) \\ f : & (M_{t})_{t≥0} ↦ M_{∞} ≡ lim_{t→∞} M_{t} \\ g : & L^{2}(F_{∞}) → M_{2} \\ g : & M_{∞} ↦ M_{t} ≡ 𝔼(M_{∞}|F_{t}) \end{matrix}

Notice that

sup_{t} 𝔼M_{t}^{2} = ∥M_{∞}∥_{2}^{2} = 𝔼(M_{∞}^{2}) < ∞,

as M_{t} is a square integrable martingale. As L^{2}(F_{∞}) is a Hilbert space, M_{2} inherits this structure.

To see that M_{2}^{c} is a closed subspace of M_{2}, consider a Cauchy sequence {M^{(n)}} in M_{2}, equivalently {M_{∞}^{(n)}} is Cauchy in L^{2}(F_{∞}). Hence M_{∞}^{(n)} converges to a limit, M_{∞} say, in L^{2}(F_{∞}). Let M_{t} := 𝔼(M_{∞}|F_{t}), then

sup_{t≥0} |M_{t}^{(n)} − M_{t}| → 0, in L^{2},

that is M^{(n)} → M uniformly in L^{2}. Hence there exists a subsequence n(k) such that M^{n(k)} → M uniformly; as a uniform limit of continuous functions is continuous, M ∈ M_{2}^{c}. Thus M_{2}^{c} is a closed subspace of M.

5. Basics

5.1. Local Martingales

A martingale has already been defined, but a weaker definition will prove useful for stochastic calculus. Note that I’ll often drop references to the filtration F_{t}, but this nevertheless forms an essential part of the (local) martingale.

Just before we dive in and define a Local Martingale, maybe we should pause and consider the reason for considering them. The important property of local martingales will only be seen later in the notes; and as we frequently see in this subject it is one of stability that is, they are a class of objects which are closed under an operation, in this case under the stochastic integral – an integral of a previsible process with a local martingale integrator is a local martingale. Definition 5.1. M = {M_{t}, F_{t}, 0 ≤ t ≤ ∞} is a local martingale if and only if there exists a sequence of stopping times T_{n} tending to infinity such that M^{T_{n}} are martingales for all n. The space of local martingales is denotes M_{loc}, and the subspace of continuous local martingales is denotes M_{loc}^{c}.

Recall that a martingale (X_{t})_{t≥0} is said to be bounded if there exists a universal constant K such that for all ω and t ≥ 0, then |X_{t}(ω)| < K. Theorem 5.2. Every bounded local martingale is a martingale. Proof Let T_{n} be a sequence of stopping times as in the definition of a local martingale. This sequence tends to infinity, so pointwise X_{t}^{T_{n}}(ω) → X_{t}(ω). Using the conditional form of the dominated convergence theorem (using the constant bound as the dominating function), for t ≥ s ≥ 0

lim_{n→∞} 𝔼(X_{t}^{T_{n}}|F_{s}) = 𝔼(X_{t}|F_{s}).

But as X^{T_{n}} is a (genuine) martingale, 𝔼(X_{t}^{T_{n}}|F_{s}) = X_{s}^{T_{n}} = X_{T_{n}∧s}; so

𝔼(X_{t}|F_{s}) = lim_{n→∞} 𝔼(X_{t}^{T_{n}}|F_{s}) = lim_{n→∞} X_{s}^{T_{n}} = X_{s}.

Hence X_{t} is a genuine martingale.

Proposition 5.3. The following are equivalent

(i) M = {M_{t}, F_{t}, 0 ≤ t ≤ ∞} is a continuous martingale. (ii) M = {M_{t}, F_{t}, 0 ≤ t ≤ ∞} is a continuous local martingale and for all t ≥ 0, the set {M_{T} : T a stopping time, T ≤ t} is uniformly integrable. Proof

(i) ⇒ (ii) By optional stopping theorem, if T ≤ t then M_{T} = 𝔼(M_{t}|F_{T}) hence the set is uniformly integrable.

(ii) ⇒ (i)It is required to prove that 𝔼(M_{0}) = 𝔼(M_{T}) for any bounded stopping time T. Then by local martingale property for any n,

𝔼(M_{0}) = 𝔼(M_{T ∧T_{n}}),

uniform integrability then implies that

lim_{n→∞} 𝔼(M_{T ∧T_{n}}) = 𝔼(M_{T}).

5.2. Local Martingales which are not Martingales

There do exist local martingales which are not themselves martingales. The following is an example Let B_{t} be a d dimensional Brownian Motion starting from x. It can be shown using Itô’s formula that a harmonic function of a Brownian motion is a local martingale (this is on the example sheet). From standard PDE theory it is known that for d ≥ 3, the function

f(x) = \frac{1}{|x|^{d−2}}

is a harmonic function, hence X_{t} = 1/|B_{t}|^{d−2} is a local martingale. Now consider the L^{p} norm of this local martingale

𝔼_{x}|X_{t}|^{p} = ∫ \frac{1}{(2πt)^{d/2}} exp (−\frac{|y − x|^{2}}{2t} ) |y|^{−(d−2)p}dy.

Consider when this integral converges. There are no divergence problems for |y| large, the potential problem lies in the vicinity of the origin. Here the term

\frac{1}{(2πt)^{d/2}} exp (−\frac{|y − x|^{2}}{2t} )

is bounded, so we only need to consider the remainder of the integrand integrated over a ball of unit radius about the origin which is bounded by

C ∫_{B(0,1)} |y|^{−(d−2)p}dy,

for some constant C, which on tranformation into polar co-ordinates yields a bound of the form

C′ ∫_{0}^{1} r^{−(d−2)p}r^{d−1}dr,

with C′ another constant. This is finite if and only if −(d − 2)p + (d − 1) > −1 (standard integrals of the form 1/r^{k}). This in turn requires that p < d/(d − 2). So clealry 𝔼_{x}|X_{t}| will be finite for all d ≥ 3.

Now although 𝔼_{x}|X_{t}| < ∞ and X_{t} is a local martingale, we shall show that it is not a martingale. Note that (B_{t} − x) has the same distribution as \sqrt{t}(B_{1} − x) under ℙ_{x} (the probability measure induced by the BM starting from x). So as t → ∞, |B_{t}| → ∞ in probability and X_{t} → 0 in probability. As X_{t} ≥ 0, we see that 𝔼_{x}(X_{t}) = 𝔼_{x}|X_{t}| < ∞. Now note that for any R < ∞, we can construct a bound

𝔼_{x}X_{t} ≤ \frac{1}{(2πt)^{d/2}} ∫_{|y|≤R} |y|^{−(d−2)}dy + R^{−(d−2)},

which converges, and hence

lim sup_{t→∞} 𝔼_{x}X_{t} ≤ R^{−(d−2)}.

As R was chosen arbitrarily we see that 𝔼_{x}X_{t} → 0. But 𝔼_{x}X_{0} = |x|^{−(d−2)} > 0, which implies that 𝔼_{x}X_{t} is not constant, and hence X_{t} is not a martingale.

6. Total Variation and the Stieltjes Integral

Let A : [0, ∞) → ℝ be a CADLAG (continuous to right, with left limits) process. Let a partition Π = {t_{0}, t_{1}, …, t_{m}} have 0 = t_{0} ≤ t_{1} ≤ ⋯ ≤ t_{m} = t; the mesh of the partition is defined by

δ(Π) = max_{1≤k≤m} |t_{k} − t_{k−1}|.

The variation of A is then defined as the increasing process V given by,

V_{t} := sup_{Π} { ∑_{k=1}^{n(Π)} |A_{t_{k}∧t} − A_{t_{k−1}∧t}| : 0 = t_{0} ≤ t_{1} ≤ ⋯ ≤ t_{n} = t} .

An alternative definition is given by

V_{t}^{0} := lim_{n→∞} ∑_{1}^{n(Π)} |A_{k2^{−n}∧t} − A_{(k−1)2^{−n}∧t}| .

These can be shown to be equivalent (for CADLAG processes), since trivially (use the dyadic partition), V_{t}^{0} ≤ V_{t}. It is also possible to show that V_{t}^{0} ≥ V_{t} for the total variation of a CADLAG process. Definition 6.1. A process A is said to have finite variation if the associated variation process V is finite (i.e. if for every t and every ω, |V_{t}(ω)| < ∞.

6.1. Why we need a Stochastic Integral

Before delving into the depths of the integral it’s worth stepping back for a moment to see why the ‘ordinary’ integral cannot be used on a path at a time basis (i.e. separately for each ω ∈ Ω). Suppose we were to do this i.e. set

I_{t}(X) = ∫_{0}^{t} X_{s}(ω)dM_{s}(ω),

for M ∈ M_{2}^{c}; but for an interesting martingale (i.e. one which isn’t zero a.s.), the total variation is not finite, even on a bounded interval like [0, T]. Thus the Lebesgue-Stieltjes integral definition isn’t valid in this case. To generalise we shall see that the quadratic variation is actually the ‘right’ variation to use (higher variations turn out to be zero and lower ones infinite, which is easy to prove by considering the variation expressed as the limit of a sum and factoring it by a maximum multiplies by the quadratic variation, the first term of which tends to zero by continuity). But to start, we shall consider integrating a previsible process H_{t} with an integrator which is an increasing finite variation process. First we shall prove that a continuous local martingale of finite variation is zero.

Proposition 6.2. If M is a continuous local martingale of finite variation, starting from zero then M is identically zero. Proof Let V be the variation process of M. This V is a continuous, adapted process. Now define a sequence of stopping times S_{n} as the first time V exceeds n, i.e. S_{n} := inf_{t}{t ≥ 0 : V_{t} ≥ n}. Then the martingale M^{S_{n}} is of bounded variation. It therefore suffices to prove the result for a bounded, continuous martingale M of bounded variation.

Fix t ≥ 0 and let {0 = t_{0}, t_{1}, …, t_{N} = t} be a partition of [0, t]. Then since M_{0} = 0 it is clear that, M_{t}^{2} = ∑_{k=1}^{N} (M_{t_{k}}^{2} − M_{t_{k−1}}^{2}). Then via orthogonality of martingale increments

\begin{matrix}𝔼(M_{t}^{2}) = & 𝔼 (∑_{k=1}^{N} (M_{t_{k}} − M_{t_{k−1}})^{2}) \\ ≤ & 𝔼 (V_{t} sup_{k} |M_{t_{k}} − M_{t_{k−1}} |) \end{matrix}

The integrand is bounded by n^{2} (from definition of the stopping time S_{n}), hence the expectation converges to zero as the modulus of the partition tends to zero by the bounded convergence theorem. Hence M ≡ 0.

6.2. Previsibility

The term previsible has crept into the discussion earlier. Now is the time for a proper definition. Definition 6.3. The previsible (or predictable) σ-field P is the σ-field on ℝ^{+}×Ω generated by the processes (X_{t})_{t≥0}, adapted to F_{t}, with left continuous paths on (0, ∞).

Remark The same σ-field is generated by left continuous, right limits processes (i.e. càglàd processes) which are adapted to F_{t−}, or indeed continuous processes (X_{t})_{t≥0} which are adapted to F_{t−}. It is gnerated by sets of the form A × (s, t] where A ∈ F_{s}. It should be noted that càdlàg processes generate the optional σ field which is usually different. Theorem 6.4. The previsible σ fieldis also generated by the collection of random sets A × {0} where A ∈ F_{0} and A × (s, t] where A ∈ F_{s}. Proof Let the σ field generated by the above collection of sets be denotes P′. We shall show P = P′. Let X be a left continuous process, define for n ∈ N

X^{n} = X_{0}1_{0}(t) + ∑_{k} X_{k/2^{n}}1_{(k/2^{n},(k+1)/2^{n}]}(t)

It is clear that X_{n} ∈ P′. As X is left continuous, the above sequence of left-continuous processes converges pointwise to X, so X is P′ measurable, thus P ⊂ P′. Conversely consider the indicator function of A × (s, t] this can be written as 1_{[0,t_{A}]\[0,s_{A}]}, where s_{A}(ω) = s for ω ∈ A and +∞ otherwise. These indicator functions are adapated and left continuous, hence P′ ⊂ P.

Definition 6.5. A process (X_{t})_{t≥0} is said to be previsible, if the mapping (t, ω) ↦ X_{t}(ω) is measurable with respect to the previsible σ-field P.

6.3. Lebesgue-Stieltjes Integral

[In the lecture notes for this course, the Lebesgue-Stieltjes integral is considered first for functions A and H; here I consider processes on a pathwise basis.]

Let A be an increasing cadlag process. This induces a Borel measure dA on (0, ∞) such that

dA((s, t])(ω) = A_{t}(ω) − A_{s}(ω).

Let H be a previsible process (as defined above). The Lebesgue-Stieltjes integral of H is defined with respect to an increasing process A by

(H · A)_{t}(ω) = ∫_{0}^{t} H_{s}(ω)dA_{s}(ω),

whenever H ≥ 0 or (|H| · A)_{t} < ∞.

As a notational aside, we shall write

(H · A)_{t} ≡ ∫_{0}^{t} HdX,

and later on we shall use

d(H · X) ≡ HdX.

This definition may be extended to integrator of finite variation which are not increasing, by decomposing the process A of finite variation into a difference of two increasing processes, so A = A^{+} − A^{−}, where A^{±} = (V ± A)/2 (here V is the total variation process for A). The integral of H with respect to the finite variation process A is then defined by

(H · A)_{t}(ω) := (H · A^{+})_{t}(ω) − (H · A^{−})_{t}(ω),

whenever (|H| · V)_{t} < ∞.

There are no really new concepts of the integral in the foregoing; it is basically the Lebesgue-Stieltjes integral eextended from functions H(t) to processes in a pathwise fashion (that’s why ω has been included in those definitions as a reminder). Theorem 6.6. If X is a non-negative continuous local martingale and 𝔼(X_{0}) < ∞ then X_{t} is a supermartingale. If additionally X has constant mean, i.e. 𝔼(X_{t}) = 𝔼(X_{0}) for all t then X_{t} is a martingale.

Proof As X_{t} is a continuous local martingale there is a sequence of stopping times T_{n} ↑ ∞ such that X^{T_{n}} is a genuine martingale. From this martingale property

𝔼(X_{t}^{T_{n}}|F_{s}) = X_{s}^{T_{n}}.

As X_{t} ≥ 0 we can apply the conditional form of Fatou’s lemma, so

𝔼(X_{t}|F_{s}) = 𝔼(lim inf_{n→∞} X_{t}^{T_{n}}|F_{s}) ≤ lim inf_{n→∞} 𝔼(X_{t}^{T_{n}}|F_{s}) = lim inf_{n→∞} X_{s}^{T_{n}} = X_{s}.

Hence 𝔼(X_{t}|F_{s}) ≤ X_{s}, so X_{t} is a supermartingale.

Given the constant mean property 𝔼(X_{t}) = 𝔼(X_{s}). Let

A_{n} := {ω : X_{s} − 𝔼(X_{t}|F_{s}) > 1/n},

so

A := ⋃_{n=1}^{∞} A_{n} = {ω : X_{s} − 𝔼(X_{t}|F_{s}) > 0}.

Consider ℙ(A) = ℙ(∪_{n=1}^{∞}A_{n}) ≤ ∑_{n=1}^{∞} ℙ(A_{n}). Suppose for some n, ℙ(A_{n}) > ϵ, then note that

\begin{matrix}ω ∈ A_{n} : & X_{s} − 𝔼(X_{t}|F_{s}) > 1/n \\ ω ∈ Ω/A_{n} : & X_{s} − 𝔼(X_{t}|F_{s}) ≥ 0 \end{matrix}

Hence

X_{s} − 𝔼(X_{t}|F_{s}) ≥ \frac{1}{n}1_{A_{n}},

taking expectations yields

𝔼(X_{s}) − 𝔼(X_{t}) > \frac{ϵ}{n},

but by the constant mean property the left hand side is zero; hence a contradiction, thus all the ℙ(A_{n}) are zero, so

X_{s} = 𝔼(X_{t}|F_{s}) a.s.

7. The Integral

We would like eventually to extend the definition of the integral to integrands which are previsible processes and integrators which are semimartingales (to be defined later in these notes). In fact in these notes we’ll only get as far as continuous semimartingales; but it is possible to go the whole way and define the integral of a previsible process with respect to a general semimartingale; but some extra problems are thrown up on the way, in particular as regards the construction of the quadratic variation process of a discontinuous process.

Various special classes of process will be needed in the sequel and these are all defined here for convenience. Naturally with terms like ‘elementary’ and ‘simple’ occurring many books have different names for the same concepts – so beware!

7.1. Elementary Processes

An elementary process H_{t}(ω) is one of the form

H_{t}(ω) = Z(ω)1_{(S(ω),T (ω)]}(t),

where S, T are stopping times, S ≤ T ≤ ∞, and Z is a bounded F_{S} measurable random variable.

Such a process is the simplest non-trivial example of a previsible process. Let’s prove that it is previsible:

H is clearly a left continuous process, so we need only show that it is adapted. It can be considered as the pointwise limit of a sequence of right continuous processes

H_{n}(t) = lim_{n→∞} Z1_{[S_{n},T_{n})},  S_{n} = S + \frac{1}{n},  T_{n} = T + \frac{1}{n}.

So it is sufficient to show that Z1_{[U,V)} is adapted when U and V are stopping times which satisfy U ≤ V, and Z is a bounded F_{U} measurable function. Let B be a borel set of ℝ, then the event

{Z1_{[U,V)}(t) ∈ B} = [{Z ∈ B} ∩ {U ≤ t}] ∩ {V > t}.

By the definition of U as a stopping time and hence the definition of F_{U}, the event enclosed by square brackets is in F_{t}, and since V is a stopping time {V > t} = Ω/{V ≤ t} is also in F_{t}; hence Z1_{[U,V)} is adapted.

7.2. Strictly Simple and Simple Processes

A process H is strictly simple (H ∈ L^{∗}) if there exist 0 ≤ t_{0} ≤ ⋯ ≤ t_{n} < ∞ and uniformly bounded F_{t_{k}} measurable random variables Z_{k} such that

H = H_{0}(ω)1_{0}(t) ∑_{k=0}^{n−1} Z_{k}(ω)1_{(t_{k},t_{k+1}](t)}.

This can be extended to H is a simple processes (H ∈ L), if there exists a sequence of stopping times 0 ≤ T_{0} ≤ ⋯ ≤ T_{k} → ∞, and Z_{k} uniformly bounded F_{T_{k}} measurable random variables such that

H = H_{0}(ω)1_{0}(t) + ∑_{k=0}^{∞} Z_{k}1_{(T_{k},T_{k+1}]}.

Similarly a simple process is also a previsible process. The fundamental result will follow from the fact that the σ-algebra generated by the simple processes is exactly the previsible σ-algebra. We shall see the application of this after the next section.

8. The Stochastic Integral

As has been hinted at earlier the stochastic integral must be built up in stages, and to start with we shall consider integrators which are L^{2} bounded martingales, and integrands which are simple processes.

8.1. Integral for H ∈ L and M ∈ M_{2}

For a simple process H ∈ L, and M an L^{2} bounded martingale then the integral may be defined by the ‘martingale transform’ (c.f. discrete martingale theory)

∫_{0}^{t} H_{s}dM_{s} = (H · M)_{t} := ∑_{k=0}^{∞} Z_{k} (M_{T_{k+1}∧t} − M_{T_{k}∧t})

Proposition 8.1. If H is a simple process, M a L_{2} bounded martingale, and T a stopping time. Then

(i) (H · M)^{T} = (H1_{(0,T]}) · M = H · (M^{T}). (ii) (H · M) ∈ M_{2}. (iii) E[(H · M)_{∞}^{2}] = ∑_{k=0}^{∞} 𝔼[Z_{k}^{2}(M_{T_{k+1}}^{2} − M_{T_{k}}^{2})] ≤ ∥H∥_{∞}^{2}𝔼(M_{∞}^{2}). Proof

Part (i) As H ∈ L we can write

H = ∑_{k=0}^{∞} Z_{k}1_{(T_{k},T_{k+1}]},

for T_{k} stopping times, and Z_{k} an F_{T_{k}} measurable bounded random variable. By our definition for M ∈ M^{2}, we have

(H · M)_{t} = ∑_{k=0}^{∞} Z_{k} (M_{T_{k+1}∧t} − M_{T_{k}∧t}) ,

and so, for T a general stopping time consider (H · M)_{t}^{T} = (H · M)_{T ∧t} and so

(H · M)_{t}^{T} = ∑_{k=0}^{∞} Z_{k} (M_{T_{k+1}∧T ∧t} − M_{T_{k}∧T ∧t}) .

Similar computations can be performed for (H · M^{T}), noting that M_{t}^{T} = M_{T ∧t} and for (H1_{(0,T]} · M) yielding the same result in both cases. Hence

(H · M)^{T} = (H1_{(0,T]} · M) = (H · M^{T}).

Part (ii) To prove this result, first we shall establish it for an elementary process H ∈ E, and then extend to L by linearity. Suppose

H = Z1_{(R,S]},

where R and S are stopping times and Z is a bounded F_{R} measurable random variable. Let T be an arbitrary stopping time. We shall prove that

𝔼 ((H · M)_{T}) = 𝔼 ((H · M)_{0}) ,

and hence via optional stopping conclude that (H · M)_{t} is a martingale.

Note that

(H · M)_{∞} = Z (M_{S} − M_{R}) ,

and hence as M is a martingale, and Z is F_{R} measurable we obtain

\begin{matrix}𝔼(H · M)_{∞} = & 𝔼 (𝔼 (Z (M_{S} − M_{R})) |F_{R}) = 𝔼 (Z𝔼 ((M_{S} − M_{R}) |F_{R})) \\ = & 0. \end{matrix}

Via part (i) note that 𝔼(H · M)_{T} = 𝔼(H · M^{T}), so

𝔼(H · M)_{T} = 𝔼(H · M^{T})_{∞} = 0.

Thus (H · M)_{t} is a martingale by optional stopping theorem. By linearity, this result extends to H a simple process (i.e. H ∈ L).

Part (iii) We wish to prove that (H ·M) is an L^{2} bounded martingale. We again start by considering H ∈ E, an elementary process, i.e.

H = Z1_{(R,S]},

where as before R and S are stopping times, and Z is a bounded F_{R} measurable random variable.

\begin{matrix}𝔼 ((H · M)_{∞}^{2}) = & 𝔼 (Z^{2}(M_{S} − M_{R})^{2}) , \\ = & 𝔼 (Z^{2}𝔼 ((M_{S} − M_{R})^{2}|F_{R})) , \end{matrix}

where Z^{2} is removed from the conditional expectation since it is and F_{R} measurable random variable. Using the same argument as used in the orthogonality of martingale increments proof,

𝔼 ((H · M)_{∞}^{2}) = 𝔼 (Z^{2}𝔼 ((M_{S}^{2} − M_{R}^{2})|F_{R})) = 𝔼 (Z^{2} (M_{S}^{2} − M_{R}^{2})) .

As M is an L^{2} bounded martingale and Z is a bounded process,

𝔼 ((H · M)_{∞}^{2}) ≤ sup_{ω∈Ω} 2|Z(ω)|^{2}𝔼 (M_{∞}^{2}) .

so (H · M) is an L^{2} bounded martingale; so together with part (ii), (H · M) ∈ M_{2}.

To extend this to simple processes is similar, but requires a little care In general the orthogonality of increments arguments extends to the case where only finitely many of the Z_{k} in the definition of the simple process H are non zero. Let K be the largest k such that Z_{k} ≢ 0.

𝔼 ((H · M)_{∞}^{2}) = ∑_{k=0}^{K} 𝔼 (Z_{k}^{2} (M_{T_{k+1}}^{2} − M_{T_{k}}^{2})) ,

which can be bounded as

\begin{matrix}𝔼 ((H · M)_{∞}^{2}) ≤ & ∥H_{∞}∥^{2}𝔼 (∑_{k=0}^{K} (M_{T_{k+1}}^{2} − M_{T_{k}}^{2})) \\ ≤ & ∥H_{∞}∥^{2}𝔼 (M_{T_{K+1}}^{2} − M_{T_{0}}^{2}) ≤ ∥H_{∞}∥^{2}𝔼M_{∞}^{2}, \end{matrix}

since we require T_{0} = 0, and M ∈ M_{2}, so the final bound is obtained via the L^{2} martingale convergence theorem.

Now extend this to the case of an infinite sum; let n ≤ m, we have that

(H · M)_{T_{m}} − (H · M)_{T_{n}} = (H1_{(T_{n},T_{m}]} · M),

applying the result just proven for finite sums to the right hand side yields

\begin{aligned}∥(H · M)_{∞}^{T_{m}} − (H · M)_{∞}^{T_{n}} ∥_{2}^{2} & = ∑_{k=n}^{m−1} 𝔼 (Z_{k}^{2} (M_{T_{k+1}}^{2} − M_{T_{k}}^{2})) \\ & ≤ ∥H_{∞}∥_{2}^{2} 𝔼 (M_{∞}^{2} − M_{T_{n}}^{2}) . \end{aligned}

But by the L^{2} martingale convergence theorem the right hand side of this bound tends to zero as n → ∞; hence (H · M)^{T_{n}} converges in M_{2} and the limit must be the pointwise limit (H · M). Let n = 0 and m → ∞ and the result of part (iii) is obtained.

8.2. Quadratic Variation

We mentioned earlier that the total variation is the variation which is used by the usual Lebesgue-Stieltjes integral, and that this cannot be used for defining a stochastic integral, since any continuous local martingale of finite variation is indistinguishable from zero. We are now going to look at a variation which will prove fundamental for the construction of the integral. All the definitions as given here aren’t based on the partition construction. This is because I shall follow Dellacherie and Meyer and show that the other definitions are equivalent by using the stochastic integral. Theorem 8.2. The quadratic variation process ⟨M⟩_{t} of a continuous L^{2} integrable martingale M is the unique process A_{t} starting from zero such that M_{t}^{2} − A_{t} is a uniformly integrable martingale.

Proof For each n define stopping times

S_{0}^{n} := 0, S_{k+1}^{n} := inf {t > S_{k}^{n} : |M_{t} − M_{S_{k}^{n}} | > 2^{−n}} for k ≥ 0

Define

T_{k}^{n} := S_{k}^{n} ∧ t

Then

\begin{matrix}M_{t}^{2} = & ∑_{k≥1} (M_{t∧S_{k}^{n}}^{2} − M_{t∧S_{k−1}^{n}}^{2}) \\ = & ∑_{k≥1} (M_{T_{k}^{n}}^{2} − M_{T_{k−1}^{n}}^{2}) \\ = & 2 ∑_{k≥1} M_{T_{k−1}^{n}} (M_{T_{k}^{n}} − M_{T_{k−1}^{n}}) + ∑_{k≥1} (M_{T_{k}^{n}} − M_{T_{k−1}^{n}})^{2} \end{matrix}  (∗)

Now define H^{n} to be the simple process given by

H^{n} := ∑_{k≥1} M_{S_{k−1}^{n}}1_{(S_{k−1}^{n},S_{k}^{n}]}.

We can then think of the first term in the decomposition (∗) as (H^{n} · M). Now define

A_{t}^{n} := ∑_{k≥1} (M_{T_{k}^{n}} − M_{T_{k−1}^{n}})^{2} ,

so the expression (∗) becomes

M_{t}^{2} = 2(H^{n} · M)_{t} + A_{t}^{n}.  (∗∗)

From the construction of the stopping times S_{k}^{n} we have the following properties

∥H^{n} − H^{n+1}∥_{∞} = sup_{t} |H_{t}^{n} − H_{t}^{n+1}| ≤ 2^{−(n+1)}
∥H^{n} − H^{n+m}∥_{∞} = sup_{t} |H_{t}^{n} − H_{t}^{n+m}| ≤ 2^{−(n+1)} for all m ≥ 1
∥H^{n} − M∥_{∞} = sup_{t} |H_{t}^{n} − M_{t}| ≤ 2^{−n}

Let J_{n}(ω) be the set of all stopping times S_{k}^{n}(ω) i.e.

J_{n}(ω) := {S_{k}^{n}(ω) : k ≥ 0}.

Clearly J_{n}(ω) ⊂ J_{n+1}(ω). Now for any m ≥ 1, using Proposition 8.1(iii) the following result holds

\begin{matrix}𝔼 ([(H^{n} · M) − (H^{n+m} · M)]_{∞}^{2}) = & 𝔼 ([(\{H^{n} − H^{n+m}\} · M)]_{∞}^{2}) \\ ≤ & ∥H^{n} − H^{n+m}∥_{∞}^{2}𝔼(M_{∞}^{2}) \\ ≤ & (2^{−(n+1)})^{2} 𝔼(M_{∞}^{2}). \end{matrix}

Thus (H^{n} · M)_{∞} is a Cauchy sequence in the complete Hilbert space L^{2}(F_{∞}); hence by completeness of the Hilbert Space it converges to a limit in the same space. As (H^{n} · M) is a continuous martingale for each n, so is the the limit N say. By Doob’s L^{2} inequality applied to the continuous martingale (H^{n} · M) − N,

𝔼 (sup_{t≥0} |(H^{n} · M) − N|^{2}) ≤ 4𝔼 ([(H · M) − N]_{∞}^{2}) →_{n→∞} 0.

Hence (H^{n} · M) converges to N uniformly a.s.. From the relation (∗∗) we see that as a consequence of this, the process A^{n} converges uniformly a.s. to a process A, where

M_{t}^{2} = 2N_{t} + A_{t}.

Now we must check that this limit process A is increasing. Clearly A^{n}(S_{k}^{n}) ≤ A^{n}(S_{k+1}^{n}), and since J_{n}(ω) ⊂ J_{n+1}(ω), it is also true that A(S_{k}^{n}) ≤ A(S_{k+1}^{n}) for all n and k, and so A is certainly increasing on the closure of J(ω) := ∪_{n}J_{n}(ω). However if I is an open interval in the complement of J, then no stopping time S_{k}^{n} lies in this interval, so M must be constant throughout I, so the same is true for the process A. Hence the process A is continuous, increasing, and null at zero; such that M_{t}^{2} − A_{t} = 2N_{t}, where N_{t} is a UI martingale (since it is L^{2} bounded). Thus we have established the existence result. It only remains to consider uniqueness.

Uniqueness follows from the result that a continuous local martingale of finite variation is everywhere zero. Suppose the process A in the above definition were not unique. That is suppose that also for some B_{t} continuous increasing from zero, M_{t}^{2} −B_{t} is a UI martingale. Then as M_{t}^{2} − A_{t} is also a UI martingale by subtracting these two equations we get that A_{t} − B_{t} is a UI martingale, null at zero. It clearly must have finite variation, and hence be zero.

The following corollary will be needed to prove the integration by parts formula, and can be skipped on a first reading; however it is clearer to place it here, since this avoids having to redefine the notation. Corollary 8.3. Let M be a bounded continuous martingale, starting from zero. Then

M_{t}^{2} = 2 ∫_{0}^{t} M_{s}dM_{s} + ⟨M⟩_{t}.

Proof In the construction of the quadratic variation process the quadratic variation was constructed as the uniform limit in L^{2} of processes A_{t}^{n} such that

A_{t}^{n} = M_{t}^{2} − 2(H^{n} · M)_{t},

where each H^{n} was a bounded previsible process, such that

sup_{t} |H_{t}^{n} − M| ≤ 2^{−n},

and hence H^{n} → M in L^{2}(M), so the martingales (H^{n} ·M) converge to (M ·M) uniformly in L^{2}, hence it follows immediately that

M_{t}^{2} = 2 ∫_{0}^{t} M_{s}dM_{s} + ⟨M⟩_{t},

Theorem 8.4. The quadratic variation process ⟨M⟩_{t} of a continuous local martingale M is the unique increasing process A, starting from zero such that M^{2} − A is a local martingale. Proof We shall use a localisation technique to extend the definition of quadratic variation from L^{2} bounded martingales to general local martingales.

The mysterious seeming technique of localisation isn’t really that complex to understand. The idea is that it enables us to extend a definition which applies for ‘X widgets’ to one valid for ‘local X widgets’. It achieves this by using a sequence of stopping times which reduce the ‘local X widgets’ to ‘X widgets’ ; the original definition can then be applied to the stopped version of the ‘X widget’. We only need to check that we can sew up the pieces without any holes i.e. that our definition is independent of the choice of stopping times!

Let T_{n} = inf{t : |M_{t}| > n}, define a sequence of stopping times. Now define

⟨M⟩_{t} := ⟨M^{T_{n}}⟩ for 0 ≤ t ≤ T_{n}

To check the consistency of this definition note that

⟨M^{T_{n}}⟩^{T_{n−1}} = ⟨M^{T_{n−1}}⟩

and since the sequence of stopping times T_{n} → ∞, we see that ⟨M⟩ is defined for all t. Uniqueness follows from the result that any finite variation continuous local martingale starting from zero is identically zero.

The quadratic variation turns out to be the ‘right’ sort of variation to consider for a martingale; since we have already shown that all but the zero martingale have infinite total variation; and it can be shown that the higher order variations of a martingale are zero a.s.. Note that the definition given is for a continuous local martingale; we shall see later how to extend this to a continuous semimartingale.

8.3. Covariation

From the definition of the quadratic variation of a local martingale we can define the covariation of two local martingales N and M which are locally L^{2} bounded via the polarisation identity

⟨M, N⟩ := \frac{⟨M + N⟩ − ⟨M − N⟩}{4}.

We need to generalise this slightly, since the above definition required the quadratic variation terms to be finite. We can prove the following theorem in a straightforward manner using the definition of quadratic variation above, and this will motivate the general definition of the covariation process.

Theorem 8.5. For M and N two local martingales which are locally L^{2} bounded then there exists a unique finite variation process A starting from zero such that MN − A is a local martingale. This process A is the covariation of M and N.

This theorem is turned round to give the usual definition of the covariation process of two continuous local martingales as: Definition 8.6. For two continuous local martingales N and M, there exists a unique finite variation process A, such that MN − A is a local martingale. The covariance process of N and M is defined as this process A.

It can readily be verified that the covariation process can be regarded as a symmetric bilinear form on the space of local martingales, i.e. for L,M and N continuous local martingales

\begin{matrix}⟨M + N, L⟩ = & ⟨M, L⟩ + ⟨N, L⟩, \\ ⟨M, N⟩ = & ⟨N, M⟩, \\ ⟨λM, N⟩ = & λ⟨M, N⟩, λ ∈ ℝ. \end{matrix}

8.4. Extension of the Integral to L^{2}(M)

We have previously defined the integral for H a simple process (in L), and M ∈ M_{2}^{c}, and we have noted that (H · M) is itself in M_{2}. Hence

𝔼 ((H · M)_{∞}^{2}) = 𝔼 (Z_{i−1}^{2} (M_{T_{i}} − M_{T_{i−1}})^{2})

Recall that for M ∈ M_{2}, then M^{2} − ⟨M⟩ is a uniformly integrable martingale. Hence for S and T stopping times such that S ≤ T, then

𝔼 ((M_{T} − M_{S})^{2}|F_{S}) = 𝔼(M_{T}^{2} − M_{S}^{2}|F_{S}) = 𝔼(⟨M⟩_{T} − ⟨M⟩_{S}|F_{S}).

So summing we obtain

\begin{matrix}𝔼 ((H · M)_{∞}^{2}) = & 𝔼 ∑ Z_{i−1}^{2} (⟨M⟩_{T_{i}} − ⟨M⟩_{T_{i−1}}) , \\ = & 𝔼 ((H^{2} · ⟨M⟩)_{∞}) . \end{matrix}

In the light of this, we define a seminorm ∥H∥_{M} via

∥H∥_{M} = [𝔼 ((H^{2} · ⟨M⟩)_{∞})]^{1/2} = [𝔼 (∫_{0}^{∞} H_{s}^{2}d⟨M⟩_{s})]^{1/2}.

The space L^{2}(M) is then defined as the subspace of the previsible processes, where this seminorm is finite, i.e.

L^{2}(M) := {previsible processes H such that ∥H∥_{M} < ∞}.

However we would actually like to be able to treat this as a Hilbert space, and there remains a problem, namely that if X ∈ L^{2}(M) and ∥X∥_{M} = 0, this doesn’t imply that X is the zero process. Thus we follow the usual route of defining an equivalence relation via X ∼ Y if and only if ∥X − Y ∥_{M} = 0. We now define

L^{2}(M) := {equivalence classes of previsible processes H such that ∥H∥_{M} < ∞},

and this is a Hilbert space with norm ∥ · ∥_{M} (it can be seen that it is a Hilbert space by considering it as suitable L^{2} space).

This establishes an isometry (called the Itô isometry) between the spaces L^{2}(M) ∩ L and L^{2}(F_{∞}) given by

\begin{matrix}I : & L^{2}(M) ∩ L → L^{2}(F_{∞}) \\ I : & H ↦ (H · M)_{∞} \end{matrix}

Remember that there is a basic bijection between the space M_{2} and the Hilbert Space L^{2}(F_{∞}) in which each square integrable martingale M is represented by its limiting value M_{∞}, so the image under the isometry (H ·M)_{∞} in L^{2}(F_{∞}) may be thought of a describing an M_{2} martingale. Hence this endows M_{2} with a Hilbert Space structure, with an inner product given by

(M, N) = 𝔼 (N_{∞}M_{∞}) .

We shall now use this Itô isometry to extend the definition of the stochastic integral from L (the class of simple processes) to the whole of L^{2}(M). Roughly speaking we shall approximate an element of L^{2}(M) via a sequence of simple processes converging to it; just as in the construction of the Lebesgue Integral. In doing this, we shall use the Monotone Class Theorem.

Recall that in the conventional construction of the Lebesgue integration, and proof of the elementary results the following standard machine is repeatedly invoked. To prove a ‘linear’ result for all h ∈ L^{1}(S, Σ, μ), proceed in the following way:

(i) Show the result is true for h an indicator function.

(ii) Show that by linearity the result extends to all positive step functions. (iii) Use the Monotone convergence theorem to see that if h_{n} ↑ h, where the h_{n} are

step functions, then the result must also be true for h a non-negative, Σ measurable function.

(iv) Write h = h^{+} −h^{−} where both h^{+} and h^{−} are non-negative functions and use linearity to obtain the result for h ∈ L^{1}. The monotone class lemmas is a replacement for this procedure, which hides away all

the ‘machinery’ used in the constructions.

Monotone Class Theorem. Let A be π-system generating the σ-algebra F (i.e. σ(A) = F). If H is a linear set of bounded functions from Ω to ℝ satisfying

(i) 1_{A} ∈ H, for all A ∈ A, (ii) 0 ≤ f_{n} ↑ f, where f_{n} ∈ H and f is a bounded function f : Ω → ℝ, then this implies that f ∈ H, then H contains every bounded, F-measurable function f : Ω → ℝ.

In order to apply this in our case, we need to prove that the σ-algebra of previsible processes is that generated by the simple processes.

The Previsible σ-field and the Simple Processes It is fairly simple to show that the space of simple processes L forms a vector space (exercise: check linearity, constant multiples and zero). Lemma 8.7. The σ-algebra generated by the simple processes is the previsible σ-algebra i.e. the previsible σ-algebra us the smallest σ-algebra with respect to which every simple process is measurable. Proof It suffices to show that every left continuous right limit process, which is bounded and adapted to F_{t} is measurable with respect to the σ-algebra generated by the simple processes. Let H_{t} be a bounded left continuous right limits process, then

H = lim_{k→∞} lim_{n→∞} ∑_{i=2}^{nk} H_{(i−1)/n} (\frac{i − 1}{n}, \frac{i}{n}] ,

and if H_{t} is adapted to F_{t} then H_{(i−1)/n} is a bounded element of F_{(i−1)/n}.

We can now apply the Monotone Class Theorem to the vector space H of processes with a time parameter in (0, ∞), regarded as maps from (0, ∞) × Ω → ℝ. Then if this vector space contains all the simple processes i.e. L ⊂ H, then H contains every bounded previsible process on (0, ∞).

Assembling the Pieces Since I is an isometry it has a unique extension to the closure of

U = L^{2}(M) ∩ L,

in L^{2}(M). By the application of the monotone class lemma to H = \overline{U}, and the π-system of simple processes. We see that \overline{U} must contain every bounded previsible process; hence \overline{U} = L^{2}(M). Thus the Itô Isometry extends to a map from L^{2}(M) to L^{2}(F_{∞}).

Let us look at this result more informally. For a previsible H ∈ L^{2}(M), because of the density of L in L^{2}(M), we can find a sequence of simple processes H_{n} which converges to H, as n → ∞. We then consider I(H) as the limit of the I(H_{n}). To verify that this limit is unique, suppose that H′_{n} → H as n → ∞ also, where H′_{n} ∈ L. Note that H_{n} − H′_{n} ∈ L. Also H_{n} − H′_{n} → 0 and so ((H_{n} − H′_{n}) · M) → 0, and hence by the Itô isometry the limits lim_{n→∞}(H_{n} · M) and lim_{n→∞}(H′_{n} · M) coincide.

The following result is essential in order to extend the integral to continuous local martingales.

Proposition 8.8. For M ∈ M_{2}, for any H ∈ L^{2}(M) and for any stopping time T then

(H · M)^{T} = (H1_{(0,T]} · M) = (H · M^{T}).

Proof Consider the following linear maps in turn

\begin{matrix}f_{1} : & L^{2}(F_{∞}) → L^{2}(F_{∞}) \\ f_{1} : & Y ↦ 𝔼(Y |F_{T}) \end{matrix}

This map is a contraction on L^{2}(F_{∞}) since by conditional Jensen’s inequality

𝔼(Y_{∞}|F_{T})^{2} ≤ 𝔼(Y_{∞}^{2}|F_{T}),

and taking expectations yields

∥𝔼(Y |F_{T})∥_{2}^{2} = 𝔼 (𝔼(Y_{∞}|F_{T})^{2}) ≤ 𝔼 (𝔼(Y_{∞}^{2}|F_{T})) = 𝔼(Y_{∞}^{2}) = ∥Y ∥_{2}^{2}.

Hence f_{1} is a contraction on L^{2}(F_{∞}). Now

\begin{matrix}f_{2} : & L^{2}(M) → L^{2}(M) \\ f_{2} : & H ↦ H1_{(0,T]} \end{matrix}

Clearly from the definition of ∥·∥_{M}, and from the fact that the quadratic variation process is increasing

∥H1_{(0,T]}∥_{M} = ∫_{0}^{∞} H_{s}^{2}1_{(0,T]}d⟨M⟩_{s} = ∫_{0}^{T} H_{s}^{2}d⟨M⟩_{s} ≤ ∫_{0}^{∞} H_{s}^{2}d⟨M⟩_{s} = ∥H∥_{M} .

Hence f_{2} is a contraction on L^{2}(M). Hence if I denotes the Itô isometry then f_{1} ∘ I and I ∘ f_{2} are also contractions from L^{2}(M) to L^{2}(F_{∞}), (using the fact that I is an isometry between L^{2}(M) and L^{2}(F_{∞})).

Now introduce I^{(T)}, the stochastic integral map associated with M^{T}, i.e.

I^{(T)}(H) ≡ (H · M^{T})_{∞}.

Note that

∥I^{(T)}(H)∥_{2} = ∥H∥_{M^{T}} ≤ ∥H∥_{M}.

We have previously shown that the maps f_{1} ∘ I and I ∘ f_{2} and H ↦ I^{(T)}(H) agree on the space of simple processes by direct calculation. We note that L is dense in L^{2}(M) (from application of Monotone Class Lemma to the simple processes). Hence from the three bounds above the three maps agree on L^{2}(M).

8.5. Localisation

We’ve already met the idea of localisation in extending the definition of quadratic variation from L^{2} bounded continuous martingales to continuous local martingales. In this context a previsible process {H_{t}}_{t≥0}, is locally previsible if there exists a sequence of stopping times T_{n} → ∞ such that for all n H1_{(0,T_{n}]} is a previsible process. Fairly obviously every previsible process has this property. However if in addition we want the process H to be locally bounded we need the condition that there exists a sequence T_{n} of stopping times, tending to infinity such that H1_{(0,T_{n}]} is uniformly bounded for each n.

For the integrator (a martingale of integrable variation say), the localisation is to a local martingale, that is one which has a sequence of stopping times T_{n} → ∞ such that for all n, X^{T_{n}} is a genuine martingale.

If we can prove a result like

(H · X)^{T} = (H1_{(0,T]} · X^{T})

for H and X in their original (i.e. non-localised classes) then it is possible to extend the definition of (H · X) to the local classes.

Note firstly that for H and X local, and T_{n} a reducing sequence^{1} of stopping times for both H and X then we see that (H1_{(0,T]} · X^{T}) is defined in the existing fashion. Also note that if T = T_{n−1} we can check consistency

(H1_{(0,T_{n}]} · X^{T_{n}})^{T_{n−1}} = (H · X)^{T_{n−1}} = (H1_{(0,T_{n−1}]} · X^{T_{n−1}}).

Thus it is consistent to define (H · X)_{t} on t ∈ [0, ∞) via

(H · X)^{T_{n}} = (H1_{(0,T_{n}]} · X^{T_{n}}), ∀n.

We must check that this is well defined, viz if we choose another regularising sequence S_{n}, we get the same definition of (H · X). To see this note:

(H1_{(0,T_{n}]} · X^{T_{n}})^{S_{n}} = (H1_{(0,T_{n}∧S_{n}]} · X^{T_{n}∧S_{n}}) = (H1_{(0,S_{n}]} · X^{S_{n}})^{T_{n}},

hence the definition of (H · X)_{t} is the same if constructed from the regularising sequence S_{n} as if constructed via T_{n}.

8.6. Some Important Results

We can now extend most of our results to stochastic integrals of a previsible process H with respect to a continuous local martingale M. In fact in these notes we will never drop the continuity requirement. It can be done; but it requires considerably more work, especially with regard to the definition of the quadratic variation process.

1 The reducing sequence is the sequence of stopping times tending to infinity which makes the local

version of the object into the non-local version. We can find one such sequence, because if say {T_{n}}

reduces H and {S_{n}} reduces X then T_{n} ∧ S_{n} reduces both H and X.

Theorem 8.9. Let H be a locally bounded previsible process, and M a continuous local martingale. Let T be an arbitrary stopping time. Then:

(i) (H · M)^{T} = (H1_{(0,T]} · M) = (H · M^{T}) (ii) (H · M) is a continuous local martingale (iii) ⟨H · M⟩ = H^{2} · ⟨M⟩ (iv) H · (K · M) = (HK) · M Proof The proof of parts (i) and (ii) follows from the result used in the localisation that:

(H · M)^{T} = (H1_{(0,T]} · M) = (H · M^{T})

for H bounded previsible process in L^{2}(M) and M an L^{2} bounded martingale. Using this result it suffices to prove (iii) and (iv) where M, H and K are uniformly bounded (via localisation).

Part (iii)

\begin{matrix}𝔼 [(H · M)_{T}^{2} ] = & 𝔼 [(H1_{(0,T]} · M) · M)_{∞}^{2}] \\ = & 𝔼 [(H1_{(0,T]} · ⟨M⟩)_{∞}^{2}] \\ = & 𝔼 [(H^{2} · ⟨M⟩)_{T} ] \end{matrix}

Hence we see that (H ·M)^{2}−(H^{2}·⟨M⟩) is a martingale (via the optional stopping theorem), and so by uniqueness of the quadratic variation process, we have established

⟨H · M⟩ = H^{2} · ⟨M⟩.

Part (iv) The truth of this statement is readily established for H and K simple processes (in L). To extend to H and K bounded previsible processes note that

\begin{matrix}𝔼 [(H · (K · M))_{∞}^{2}] = & 𝔼 [(H^{2} · ⟨K · M⟩)_{∞}] \\ = & 𝔼 [(H^{2} · (K^{2} · ⟨M⟩))_{∞}] \\ = & 𝔼 [((HK)^{2} · ⟨M⟩)_{∞}] \\ = & 𝔼 [((HK) · M)_{∞}^{2}] \end{matrix}

Also note the following bound

𝔼 [(HK)^{2} · ⟨M⟩)_{∞}] ≤ min {∥H∥_{∞}^{2}∥K∥_{M}^{2}, ∥H∥_{M}^{2}∥K∥_{∞}^{2}} .

9. Semimartingales

I mentioned at the start of these notes that the most general form of the stochastic integral would have a previsible process as the integrand and a semimartingale as an integrator. Now it’s time to extend the definition of the Itô integral to the case of semimartingale integrators. Definition 9.1. A process X is a semimartingale if X is an adapted CADLAG process which has a decomposition

X = X_{0} + M + A,

where M is a local martingale, null at zero and A is a process null at zero, with paths of finite variation.

Note that the decomposition is not necessarily unique as there exist martingales which have finite variation. To remove many of these difficulties we shall impose a continuity condition, since under this most of our problems will vanish. Definition 9.2. A continuous semimartingale is a process (X_{t})_{t≥0} which has a Doob-Meyer decomposition

X = X_{0} + M + A,

where X_{0} is F_{0}-measurable, M_{0} = A_{0} = 0, M_{t} is a continuous local martingale and A_{t} is a continuous adapted process of finite variation. Theorem 9.3. The Doob-Meyer decomposition in the definition of a continuous semimartingale is unique. Proof Let another such decomposition be

X = X_{0} + M′ + A′,

where M′ is a continuous local martingale and A a continuous adapted process of finite variation. Then consider the process N, where

N = M′ − M = A′ − A,

by the first equality, N is the difference of two continuous local martingales, and hence is itself a continuous local martingale; and by the second inequality it has finite variation. Hence by an earlier proposition (5.2) it must be zero. Hence M′ = M and A′ = A.

We define† the quadratic variation of the continuous semimartingale as that of the continuous local martingale part i.e. for X = X_{0} + M + A,

⟨X⟩ := ⟨M⟩.

† These definitions can be made to look natural by considering the quadratic variation defined in terms

of a sum of squared increments; but following this approach, these are result which are proved later

using the Itô integral, since this provided a better approach to the discontinuous theory.

Similarly if Y + Y_{0} + N + B is another semimartingale, where B is finite variation and N is a continuous local martingale, we define

⟨X, Y ⟩ := ⟨M, N⟩.

We can extend the definition of the stochastic integral to continuous semimartingale integrators by defining

(H · X) := (H · M) + (H · A),

where the first integral is a stochastic integral as defined earlier and the second is a Lebesgue-Stieltjes integral (as the integrator is a process of finite variation).

10. Relations to Sums

This section is optional; and is included to bring together the two approaches to the constructions involved in the stochastic integral.

For example the quadratic variation of a process can either be defined in terms of martingale properties, or alternatively in terms of sums of squares of increments.

10.1. The UCP topology

We shall meet the notion of convergence uniformly on compacts in probability when considering stochastic integrals as limits of sums, so it makes sense to review this topology here.

Definition 10.1. A sequence {H_{n}}_{n≥1} converges to a process H uniformly on compacts in probability (abbreviated u.c.p.) if for each t > 0,

sup_{0≤s≤t} |H_{s}^{n} − H_{s}| → 0 in probability.

At first sight this may seem to be quite an esoteric definition; in fact it is a natural extension of convergence in probability to processes. It would also appear to be quite difficult to handle, however Doob’s martingale inequalities provide the key to handling it. Let

H_{t}^{∗} = sup_{0≤s≤t} |H_{s}|,

then for Y^{n} a CADLAG process, Y^{n} converges to Y u.c.p. iff (Y^{n} − Y)^{∗} converges to zero in probability for each t ≥ 0. Thus to prove that a sequence converges u.c.p. it often suffices to apply Doob’s inequality to prove that the supremum converges to zero in L^{2}, whence it must converge to zero in probability, whence u.c.p. convergence follows.

The space of CADLAG processes with u.c.p. topology is in fact metrizable, a compatible metric is given by

d(X, Y) = ∑_{n=1}^{∞} \frac{1}{2^{n}} 𝔼 (min(1, (X − Y)_{n}^{∗})) ,

for X and Y CADLAG processes. The metric space can also be shown to be complete. For details see Protter.

Since we have just met a new kind of convergence, it is helpful to recall the other usual types of convergence on a probability space. For convenience here are the usual definitions:

Pointwise A sequence of random variables X_{n} converges to X pointwise if for all ω not in some null set,

X_{n}(ω) → X(ω).

Probability A sequence of r.v.s X_{n} converges to X in probability, if for any ϵ > 0,

ℙ (|X_{n} − X| > ϵ) → 0, as n → ∞.

L^{p} convergence A sequence of random variables X_{n} converges to X in L^{p}, if

𝔼|X_{n} − X|^{p} → 0, as n → ∞.

It is trivial to see that pointwise convergence implies convergence in probability. It is also true that L^{p} convergence implies convergence in probability as the following theorem shows Theorem 10.2. If X_{n} converges to X in L^{p} for p > 0, then X_{n} converges to X in probability. Proof Apply Chebyshev’s inequality to f(x) = x^{p}, which yields for any ϵ > 0,

ℙ (|X_{n}| ≥ ϵ) ≤ ϵ^{−p}𝔼 (|X_{n}|^{p}) → 0, as n → ∞.

Theorem 10.3. If X_{n} → X in probability, then there exists a subsequence n_{k} such that X_{n_{k}} → X a.s. Theorem 10.4. If X_{n} → X a.s., then X_{n} → X in probability.

10.2. Approximation via Riemann Sums

Following Dellacherie and Meyer we shall establish the equivalence of the two constructions for the quadratic variation by the following theorem which approximates the stochastic integral via Riemann sums.

Theorem 10.2. Let X be a semimartingale, and H a locally bounded previsible CADLAG process starting from zero. Then

∫_{0}^{t} H_{s}dX_{s} = lim_{n→∞} ∑_{k=0}^{∞} H_{t∧k2^{−n}} (X_{t∧(k+1)2^{−n}} − X_{t∧k2^{−n}}) u.c.p.

Proof Let K_{s} = H_{s}1_{s≤t}, and define the following sequence of simple process approximations

K_{s}^{n} := ∑_{k=0}^{∞} H_{t∧k2^{−n}}1_{(t∧k2^{−n},t∧(k+1)2^{−n}]}(s).

Clearly this sequence K_{s}^{n} converges pointwise to K_{s}. We can decompose the semimartingale X as X = X_{0} + A_{t} + M_{t} where A_{t} is of finite variation and M_{t} is a continuous local martingale, both starting from zero. The result that

∫_{0}^{t} K_{s}^{n}dA_{s} → ∫_{0}^{t} K_{s}dA_{s}, u.c.p.

is standard from the Lebesgue-Stieltjes theory. Let T_{k} be a reducing sequence for the continuous local martingale M such that M^{T_{k}} is a bounded martingale. Also since K is locally bounded we can find a sequence of stopping times S_{k} such that K^{S_{k}} is a bounded previsible process. It therefore suffices to prove for a sequence of stopping times R_{k} such that R_{k} ↑ ∞, then

(K^{n} · M)_{s}^{R_{k}} → (K · M)_{s}^{R_{k}}, u.c.p..

By Doob’s L^{2} inequality, and the Itô isometry we have

𝔼 [((K^{n} · M) − (K · M))^{∗}]^{2} ≤4𝔼 [(K^{n} · M) − (K · M)]^{2} ,  Doob L^{2}
≤4∥K^{n} − K∥_{M}^{2},  Itô Isometry
≤4 ∫ (K_{s}^{n} − K_{s})^{2}d⟨M⟩_{s}

As |K^{n} −K| → 0 pointwise, and K is bounded, clearly |K^{n} −K| is also bounded uniformly in n. Hence by the Dominated Convergence Theorem for the Lebesgue-Stieltjes integral

∫ (K_{s}^{n} − K_{s})^{2}d⟨M⟩_{s} → 0 a.s..

Hence, we may conclude

𝔼 ([(K^{n} · M) − (K · M)]^{∗})^{2} → 0, as n → ∞.

So

[(K^{n} · M) − (K · M)]^{∗} → 0 in L^{2},

as n → ∞; but this implies that

[(K^{n} · M) − (K · M)]^{∗} → 0 in probability.

Hence

[(K^{n} · M) − (K · M)] → 0 u.c.p.

as required, and putting the two parts together yields

∫_{0}^{t} K_{s}^{n}dX_{s} → ∫_{0}^{t} K_{s}dX_{s}, u.c.p.

which is the required result.

This result can now be applied to the construction of the quadratic variation process, as illustrated by the next theorem. Theorem 10.3. The quadratic variation process ⟨X⟩_{t} is equal to the following limit in probability

⟨X⟩_{t} = lim_{n→∞} ∑_{k=0}^{∞} (X_{t∧(k+1)2^{−n}} − X_{t∧k2^{−n}})^{2} in probability.

Proof In the theorem (7.2) establishing the existence of the quadratic variation process, we noted in (∗∗) that

A_{t}^{n} = M_{t}^{2} − 2(H^{n} · M)_{t}.

Now from application of the previous theorem

2 ∫_{0}^{t} X_{s}dX_{s} = lim_{n→∞} ∑_{k=0}^{∞} X_{t∧k2^{−n}} (X_{t∧(k+1)2^{−n}} − X_{t∧k2^{−n}}) .

In addition,

X_{t}^{2} − X_{0}^{2} = ∑_{k=0}^{∞} (X_{t∧(k+1)2^{−n}}^{2} − X_{t∧k2^{−n}}^{2}) .

The difference of these two equations yields

A_{t} = X_{0}^{2} + lim_{n→∞} ∑_{k=0}^{∞} (X_{t∧(k+1)2^{−n}} − X_{t∧k2^{−n}})^{2} ,

where the limit is taken in probability. Hence the process A is increasing and positive on the rational numbers, and hence on the whole of ℝ by right continuity.

Remark The theorem can be strengthened still further by a result of Doléans-Dade to the effect that for X a continuous semimartingale

⟨X⟩_{t} = lim_{n→∞} ∑_{k=0}^{∞} (X_{t∧(k+1)2^{−n}} − X_{t∧k2^{−n}})^{2} ,

where the limit is in the strong sense in L^{1}. This result is harder to prove (essentially the uniform integrability of the sums must be proven) and this is not done here.

11. Itô’s Formula

Itô’s Formula is the analog of integration by parts in the stochastic calculus. It is also the first place where we see a major difference creep into the theory, and realise that our formalism has found a new subtlety in the subject.

More importantly, it is the fundamental weapon used to evaluate Itô integrals; we shall see some examples of this shortly.

The Itô isometry provides a clean-cut definition of the stochastic integral; however it was originally defined via the following theorem of Kunita and Watanabe. Theorem (Kunita-Watanabe Identity) 11.1. Let M ∈ M_{2} and H and K are locally bounded previsible processes. Then (H · M)_{∞} is the unique element of L^{2}(F_{∞}) such that for every N ∈ M_{2} we have:

𝔼 [(H · M)_{∞}N_{∞}] = 𝔼 [(H · ⟨M, N⟩)_{∞}]

Moreover we have

⟨(H · M), N⟩ = H · ⟨M, N⟩.

Proof Consider an elementary function H, so H = Z1_{(S,T]}, where Z is an F_{S} measurable bounded random variable, and S and T are stopping times such that S ≤ T. It is clear that

\begin{matrix}𝔼 [(H · M)_{∞}N_{∞}] = & 𝔼 [Z (M_{T} − M_{S}) N_{∞}] \\ = & 𝔼 [Z(M_{T} N_{T} − M_{S}N_{S})] \\ = & 𝔼 [M_{∞}(H · N)_{∞}] \end{matrix}

Now by linearity this can be extended to establish the result for all simple functions (in L). We finally extend to general locally bounded previsible H, by considering a sequence (provided it exists) of simple functions H^{n} such that H^{n} → H in L^{2}(M). Then there exists a subsequence n_{k} such that H^{n_{k}} converges to H is L^{2}(N). Then

\begin{matrix}𝔼((H^{n_{k}} · M)_{∞}N_{∞}) − 𝔼((H · M)_{∞}N_{∞}) = & 𝔼(((H^{n_{k}} − H) · M)N_{∞}) \\ ≤ & \sqrt{𝔼 ([(H^{n_{k}} − H) · M)]^{2})}\sqrt{𝔼(N_{∞}^{2})} \\ ≤ & \sqrt{𝔼 ([(H^{n_{k}} · M) − (H · M)]^{2})}\sqrt{𝔼(N_{∞}^{2})} \end{matrix}

By construction H^{n_{k}} → H in L^{2}(M) which means that

∥H^{n_{k}} − H∥_{M} → 0, as k → ∞.

By the Itô isometry

𝔼 [((H^{n_{k}} − H) · M)^{2}] = ∥H^{n_{k}} − H∥_{M}^{2} → 0, as k → ∞,

that is (H^{n_{k}} · M)_{∞} → (H · M)_{∞} in L^{2}. Hence as N is an L^{2} bounded martingale, the right hand side of the above expression tends to zero as k → ∞. Similarly as (H^{n_{k}} · N)_{∞} → (H · N)_{∞} in L^{2}, we see also that

𝔼 ((H^{n_{k}} · N)_{∞}M_{∞}) → 0, as k → ∞.

Hence we can pass to the limit to obtain the result for H.

To prove the second part of the theorem, we shall first show that

⟨(H · N), (K · M)⟩ + ⟨(K · N), (H · M)⟩ = 2HK⟨M, N⟩.

By polarisation

⟨M, N⟩ = \frac{⟨M + N⟩ − ⟨M − N⟩}{4},

also

HK = \frac{(H + K)^{2} − (H − K)^{2}}{4} .

Hence

2(HK · ⟨M, N⟩) = \frac{1}{8}([(H + K)^{2} − (H − K)^{2}] · \{⟨M+N⟩ − ⟨M-N⟩\}).

Now we use the result that ⟨(H · M)⟩ = (H^{2} · ⟨M⟩) which has been proved previously in theorem (7.9(iii)), to see that

\begin{aligned}2(HK · ⟨M, N⟩) = & \frac{1}{8}(⟨(H + K) · (M + N)⟩ − ⟨(H + K) · (M − N)⟩ \\ & − ⟨(H − K) · (M + N)⟩ + ⟨(H − K) · (M − N)⟩). \end{aligned}

Considering the first two terms

⟨(H + K)·(M + N)⟩ − ⟨(H + K) · (M − N)⟩ =
=⟨(H + K) · M + (H + K) · N⟩ − ⟨(H + K) · M − (H + K) · N⟩
=4⟨(H + K) · M, (H + K) · N⟩  by polarisation
=4 (⟨H · M, H · N⟩ + ⟨H · M, K · N⟩ + ⟨K · M, H · N⟩ + ⟨K · M, K · N⟩) .

Similarly for the second two terms

⟨(H − K)·(M + N)⟩ − ⟨(H − K) · (M − N)⟩ =
=⟨(H − K) · M + (H − K) · N⟩ − ⟨(H − K) · M − (H − K) · N⟩
=4⟨(H − K) · M, (H − K) · N⟩ by polarisation
=4 (⟨H · M, H · N⟩ − ⟨H · M, K · N⟩ − ⟨K · M, H · N⟩ + ⟨K · M, K · N⟩) .

Adding these two together yields

2(HK · ⟨M, N⟩) = ⟨(H · N), (K · M)⟩ + ⟨(K · N), (H · M)⟩

Putting K ≡ 1 yields

2H · ⟨M, N⟩ = ⟨H · M, N⟩ + ⟨M, H · N⟩.

So it suffices to prove that ⟨(H · M), N⟩ = ⟨M, (H · N)⟩, which is equivalent to showing that

(H · M)N − (H · N)M

is a local martingale (from the definition of covariation process). By localisation it suffices to consider M and N bounded martingales, whence we must check that for all stopping times T,

𝔼 ((H · M)_{T} N_{T}) = 𝔼 ((H · N)_{T} M_{T}) ,

but by the first part of the theorem

𝔼 ((H · M)_{∞}N_{∞}) = 𝔼 ((H · N)_{∞}M_{∞}) ,

which is sufficient to establish the result, since

\begin{matrix}(H · M)_{T} N_{T} = & (H · M)_{∞}^{T}N_{∞}^{T} \\ (H · N)_{T} M_{T} = & (H · N)_{∞}^{T}M_{∞}^{T} \end{matrix}

Corollary 11.2. Let N, M be continuous local martingales and H and K locally bounded previsible processes, then

⟨(H · N), (K · M)⟩ = (HK · ⟨N, M⟩).

Proof Note that the covariation is symmetric, hence

\begin{matrix}⟨(H · N), (K · M)⟩ = & (H · ⟨X, (K · M)⟩) \\ = & (H · ⟨(K · M), X)⟩) \\ = & (HK · ⟨M, N⟩). \end{matrix}

We can prove a stochastic calculus analogue of the usual integration by parts formula. However note that there is an extra term on the right hand side, the covariation of the processes X and Y. This is the first major difference we have seen between the Stochastic Integral and the usual Lebesgue Integral.

Before we can prove the general theorem, we need a lemma.

Lemma (Parts for Finite Variation Process and a Martingale) 11.3. Let M be a bounded continuous martingale starting from zero, and V a bounded variation process starting from zero. Then

M_{t}V_{t} = ∫_{0}^{t} M_{s}dV_{s} + ∫_{0}^{t} V_{s}dM_{s}.

Proof For n fixed, we can write

\begin{matrix}M_{t}V_{t} = & ∑_{k≥1} M_{k2^{−n}∧t} (V_{k2^{−n}∧t} − V_{(k−1)2^{−n}∧t}) + ∑_{k≥1} V_{(k−1)2^{−n}∧t} (M_{k2^{−n}∧t} − M_{(k−1)2^{−n}∧t}) \\ = & ∑_{k≥1} M_{k2^{−n}∧t} (V_{k2^{−n}∧t} − V_{(k−1)2^{−n}∧t}) + ∫_{0}^{t} H_{s}^{n}dM_{s}, \end{matrix}

where H^{n} is the previsible simple process

H_{s}^{n} = ∑_{k≥1} V_{k2^{−n}∧t}1_{((k−1)2^{−n}∧t,k2^{−n}∧t]}(s).

These H^{n} are bounded and converge to V by the continuity of V, so as n → ∞ the second term tends to

∫_{0}^{t} V_{s}dM_{s},

and by the Dominated Convergence Theorem for Lebesgue-Stieltjes integrals, the second term converges to

∫_{0}^{t} M_{s}dV_{s},

as n → ∞.

Theorem (Integration by Parts) 11.4. For X and Y continuous semimartingales, then the following holds

X_{t}Y_{t} − X_{0}Y_{0} = ∫_{0}^{t} X_{s}dY_{s} + ∫_{0}^{t} Y_{s}dX_{s} + ⟨X, Y ⟩_{t}.

Proof It is trivial to see that it suffices to prove the result for processes starting from zero. Hence let X_{t} = M_{t} + A_{t} and Y_{t} = N_{t} + B_{t} in Doob-Meyer decomposition, so N_{t} and M_{t} are continuous local martingales and A_{t} and B_{t} are finite variation processes, all starting from zero. By localisation we can consider the local martingales M and N to be bounded martingales and the FV processes A and B to have bounded variation. Hence by the usual (finite variation) theory

A_{t}B_{t} = ∫_{0}^{t} A_{s}dB_{s} + ∫_{0}^{t} B_{s}dA_{s}.

It only remains to prove for bounded martingales N and M starting from zero that

M_{t}N_{t} = ∫_{0}^{t} M_{s}dN_{s} + ∫_{0}^{t} N_{s}dM_{s} + ⟨M, N⟩_{t}.

This follows by application of polarisation to corollary (7.3) to the quadratic variation existence theorem. Hence

(M_{t} + A_{t})(N_{t} + B_{t}) =M_{t}N_{t} + M_{t}B_{t} + N_{t}A_{t} + A_{t}B_{t}
= ∫_{0}^{t} M_{s}dN_{s} + ∫_{0}^{t} N_{s}dM_{s} + ⟨M, N⟩_{t}
+ ∫_{0}^{t} M_{s}dB_{s} + ∫_{0}^{t} B_{s}dM_{s} + ∫_{0}^{t} N_{s}dA_{s} + ∫_{0}^{t} A_{s}dN_{s}
+ ∫_{0}^{t} A_{s}dB_{s} + ∫_{0}^{t} B_{s}dA_{s}
= ∫_{0}^{t}(M_{s} + A_{s})d(N_{s} + B_{s}) + ∫_{0}^{t}(N_{s} + B_{s})d(M_{s} + A_{s}) + ⟨M, N⟩.

Reflect for a moment that this theorem is implying another useful closure property of continuous semimartingales. It implies that the product of two continuous semimartingales X_{t}Y_{t} is a continuous semimartingale, since it can be written as a stochastic integrals with respect to continuous semimartingales and so it itself a continuous semimartingale. Theorem (Itô’s Formula) 11.5. Let f : ℝ^{n} → ℝ^{n} be a twice continuously differentiable function, and also let X = (X^{1},X^{2}, …, X^{n}) be a continuous semimartingale in ℝ^{n}. Then

f(X_{t}) − f(X_{0}) = ∑_{i=1}^{n} ∫_{0}^{t} \frac{∂f}{∂x^{i}} (X_{s})dX_{s}^{i} + \frac{1}{2} ∑_{i,j=1}^{n} ∫_{0}^{t} \frac{∂f}{∂x^{i}∂x^{j}} (X_{s})d⟨X^{i}, X^{j}⟩_{s}.

Proof To prove Itô’s formula; first consider the n = 1 case to simplify the notation. Then let A be the collection of C^{2} (twice differentiable) functions f : ℝ → ℝ for which it holds. Clearly A is a vector space; in fact we shall show that it is also an algebra. To do this we must check that if f and g are in A, then their product fg is also in A. Let F_{t} = f(X_{t}) and G_{t} = g(X_{t}) be the associated semimartingales. From the integration by parts formula

F_{t}G_{t} − F_{0}G_{0} = ∫_{0}^{t} F_{s}dG_{s} + ∫_{0}^{t} G_{s}dF_{s} + ⟨F_{s}, G_{s}⟩.

However since by assumption f and g are in A, Itô’s formula may be applied to them individually, so

∫_{0}^{t} F_{s}dG_{s} = ∫_{0}^{t} f(X_{s})\frac{∂f}{∂x} (X_{s})dX_{s}.

Also by the Kunita-Watanabe formula extended to continuous local martingales we have

⟨F, G⟩_{t} = ∫_{0}^{t} f′(X_{s})g′(X_{s})d⟨X, X⟩_{s}.

Thus from the integration by parts,

F_{t}G_{t} − F_{0}G_{0} = ∫_{0}^{t} F_{s}dG_{s} + ∫_{0}^{t} G_{s}dF_{s} + ∫_{0}^{t} f′(X_{s})g′(X_{s})d⟨X, X⟩_{s},
= ∫_{0}^{t} (F_{s}g′(X_{s}) + f′(X_{s})G_{s}) dX_{s}

+ \frac{1}{2} ∫_{0}^{t} (F_{s}g′′(X_{s}) + 2f′g′(X_{s}) + f′′(X_{s})G_{s}) d⟨M⟩_{s}.

So this is just what Itô’s formula states for fg and so Itô’s formula also applies to fg; hence fg ∈ A.

Since trivially f(x) = x is in A, then as A is an algebra, and a vector space this implies that A contains all polynomials. So to complete the proof, we must approximate f by polynomials (which we can do by standard functional analysis), and check that in the limit we obtain Itô’s formula.

Introduce a sequence U_{n} := inf{t : |X_{t}| + ⟨X⟩_{t} > n}. Hence {U_{n}} is a sequence of stopping times tending to infinity. Now we shall prove Itô’s formula for twice continuously differentiable f restricted to the interval [0, U_{n}], so we can consider X as a bounded martingale. Consider a polynomial sequence f_{k} approximating f, in the sense that for r = 0, 1, 2, f_{k}^{(r)} → f^{(r)} uniformly on a compact interval. We have proved that Itô’s formula holds for all polynomial, so it holds for f_{k} and hence

f_{k}(X_{t∧U_{n}}) − f_{k}(X_{0}) = ∫_{0}^{t∧U_{n}} f′(X_{s})dX_{s} + \frac{1}{2} ∫_{0}^{t∧U_{n}} f′′_{k}(X_{s})d⟨X⟩_{s}.

Let the continuous semimartingale X have Doob-Meyer decomposition

X_{t} = X_{0} + M_{t} + A_{t},

where M is a continuous local martingale and A is a finite variation process. We can rewrite the above as

f_{k}(X_{t∧U_{n}}) − f_{k}(X_{0}) = ∫_{0}^{t∧U_{n}} f′(X_{s})dM_{s} + ∫_{0}^{t∧U_{n}} f′(X_{s})dA_{s} + \frac{1}{2} ∫_{0}^{t∧U_{n}} f′′_{k}(X_{s})d⟨M⟩_{s}.

since ⟨X⟩ = ⟨M⟩. On (0, U_{n}] the process |X| is uniformly bounded by n, so for r = 0, 1, 2 from the convergence (which is uniform on the compact interval [0, U_{n}]) we obtain

sup_{|x|≤n} |f_{k}^{(r)} − f^{(r)}| → 0 as k → ∞

And from the Itô isometry we get the required convergence.

11.1. Applications of Itô’s Formula

Let B_{t} be a standard Brownian motion; the aim of this example is to establish that:

∫_{0}^{t} B_{s}dB_{s} = \frac{1}{2}B_{t}^{2} − \frac{1}{2}t.

This example gives a nice simple demonstration that all our hard work has achieved something. The result isn’t the same as that which would be given by the ‘logical’ extension of the usual integration rules.

To prove this we apply Itô’s formula to the function f(x) = x^{2}. We obtain

f(B_{t}) − f(B_{0}) = ∫_{0}^{t} \frac{∂f}{∂x} (B_{s})dB_{s} + \frac{1}{2} ∫_{0}^{t} \frac{∂^{2}f}{∂x^{2}} (B_{s})d⟨B, B⟩_{s},

noting that B_{0} = 0 for a standard Brownian Motion we see that

B_{t}^{2} = 2 ∫_{0}^{t} B_{s}dB_{s} + \frac{1}{2}2ds,

whence we derive that

∫_{0}^{t} B_{s}dB_{s} = \frac{B_{t}^{2}}{2} − \frac{t}{2}.

For those who have read the foregoing material carefully, there are grounds to complain that there are simpler ways to establish this result, notably by consideration of the definition of the quadratic variation process. However the point of this example was to show how Itô’s formula can help in the actual evaluation of stochastic integrals; not to establish a totally new result.

11.2. Exponential Martingales

Exponential martingales play an important part in the theory. Suppose X is a continuous semimartingale starting from zero. Define:

Z_{t} = exp (X_{t} − \frac{1}{2}⟨X⟩_{t}) .

This Z_{t} is called the exponential semimartingale associated with X_{t}, and it is the solution of the stochastic differential equation

dZ_{t} = Z_{t}dX_{t},

that is

Z_{t} = 1 + ∫_{0}^{t} Z_{s}dX_{s},

so clearly if X is a continuous local martingale, i.e. X ∈ M_{loc}^{c} then this implies, by the stability property of stochastic integration, that Z ∈ M_{loc}^{c} also.

Proof For existence, apply Itô’s formula to f(x) = exp(x) to obtain

d (exp(Y_{t})) = exp(Y_{t})dY_{t} + \frac{1}{2} exp(Y_{t})d⟨Y, Y ⟩_{t}.

Hence

d (exp(X_{t} − \frac{1}{2}⟨X⟩_{t})) = exp(X_{t} − \frac{1}{2}⟨X⟩_{t})d (X_{t} − \frac{1}{2}⟨X⟩_{t})

+ \frac{1}{2} exp (X_{t} − \frac{1}{2}⟨X⟩_{t}) d ⟨X_{t} − \frac{1}{2}⟨X⟩_{t}, X_{t} − \frac{1}{2}⟨X⟩_{t}⟩

= exp(X_{t} − \frac{1}{2}⟨X⟩_{t})dX_{t} − \frac{1}{2} exp (X_{t} − \frac{1}{2}⟨X⟩_{t}) d⟨X⟩_{t}

+ \frac{1}{2} exp (X_{t} − \frac{1}{2}⟨X⟩_{t}) d⟨X⟩_{t}

=Z_{t}dX_{t}

Hence Z_{t} certainly solves the equation. Now to check uniqueness, define

Y_{t} = exp (−X_{t} + \frac{1}{2}⟨X⟩_{t}) ,

we wish to show that for every solution of the Stochastic Differential Equation Z_{t}Y_{t} is a constant. By a similar application of Itô’s formula

dY_{t} = −Y_{t}dX_{t} + Y_{t}d⟨X⟩_{t},

whence by integration by parts (alternatively consider Itô applied to f(x, y) = xy),

d(Z_{t}Y_{t}) =Z_{t}dY_{t} + Y_{t}dZ_{t} + ⟨Z, Y ⟩_{t},
=Z_{t}(−Y_{t}dX_{t} + Y_{t}d⟨X⟩_{t}) + Y_{t}Z_{t}dX_{t} + (−Y_{t}Z_{t})d⟨X⟩_{t},
=0.

So Z_{t}Y_{t} is a constant, hence the unique solution of the stochastic differential equation dZ_{t} = Z_{t}dX_{t}, with Z_{0} = 1, is

Z_{t} = exp (X_{t} − \frac{1}{2}⟨X⟩_{t}) .

Example Let X_{t} = λB_{t}, for an arbitrary scalar λ. Clearly X_{t} is a continuous local martingale, so the associated exponential martingale is

M_{t} = exp (λB_{t} − \frac{1}{2}λ^{2}t) .

It is clear that the exponential semimartingale of a real valued martingale must be non-negative, and thus by application of Fatou’s lemma we can show that it is a supermartingale, thus 𝔼(M_{t}) ≤ 1 for all t. Theorem 11.6. Let M be a non-negative local martingale, such that 𝔼M_{0} < ∞ then M is a supermartingale. Proof Let T_{n} be a reducing sequence for M_{n} − M_{0}, then for t > s ≥ 0,

\begin{aligned}𝔼(M_{t∧T_{n}}|F_{s}) & = 𝔼(M_{0}|F_{s}) + 𝔼(M_{t∧T_{n}} − M_{0}|F_{s}) \\ & = M_{0} + M_{s∧T_{n}} − M_{0} = M_{s∧T_{n}}. \end{aligned}

Now by application of the conditional form of Fatou’s lemma

𝔼(M_{t}|F_{s}) = 𝔼(lim inf_{n→∞} M_{t∧T_{n}}|F_{s}) ≤ lim inf_{n→∞} 𝔼(M_{t∧T_{n}}|F_{s}) = M_{s}.

Thus M is a supermartingale as required.

Checking for a Martingale The general exponential semimartingale is useful, but in many applications, not the least of which will be Girsanov’s formula an actual Martingale will be needed. How do we go about checking if a local martingale is a martingale anyway? It will turn out that there are various methods, some of which crop up in the section on filtration. First I shall present a simple example and then prove a more general theorem. A common error is to think that it is sufficient to show that a local martingale is locally bounded in L^{2} to show that it is a martingale – this is not sufficient as should be made clear by this example!

The exponential of λB_{t} We continue our example from above. Let M_{t} = exp(λB_{t} − 1/2λ^{2}t) be the exponential semimartingale associated with a standard Brownian Motion B_{t}, starting from zero. By the previous argument we know that

M_{t} = 1 + ∫_{0}^{t} λM_{s}dB_{s},

hence M is a local martingale. Fix T a constant time, which is of course a stopping time, then B^{T} is an L^{2} bounded martingale (𝔼(B_{t}^{T})^{2} = t ∧ T ≤ T). We then show that M^{T} is in L^{2}(B^{T}) as follows

∥M^{T} ∥_{B} = 𝔼 (∫_{0}^{T} M_{s}^{2}d⟨B⟩_{s}) ,

= 𝔼 (∫_{0}^{T} M_{s}^{2}ds) ,

≤ 𝔼 (∫_{0}^{T} exp(2λB_{s})ds) ,

= ∫_{0}^{T} 𝔼 (exp(2λB_{s})) ds,
= ∫_{0}^{T} exp(2λ^{2}s^{2})ds < ∞.

In the final equality we use that fact that B_{s} is distributed as N(0, s), and we use the characteristic function of the normal distribution. Thus by the integration isometry theorem we have that (M^{T} · B)_{t} is an L^{2} bounded martingale. Thus for every such T, Z^{T} is an L^{2} bounded martingale, which implies that M is a martingale.

The Exponential Martingale Inequality We have seen a specific proof that a certain (important) exponential martingale is a true martingale, we now show a more general argument.

Theorem 11.7. Let M be a continuous local martingale, starting from zero. Suppose for each t, there exists a constant K_{t} such that ⟨M⟩_{t} < ∞ a.s., then for every t, and every y > 0,

ℙ [ sup_{0≤s≤t} M_{s} > y] ≤ exp(−y^{2}/2K_{t}).

Furthermore, the associated exponential semimartingale Z_{t} = exp(θM_{t} − 1/2θ^{2}⟨M⟩_{t}) is a true martingale.

Proof We have already noted that the exponential semimartingale Z_{t} is a supermartingale, so 𝔼(Z_{t}) ≤ 1 for all t ≥ 0, and hence for θ > 0 and y > 0,

ℙ [sup_{s≤t} M_{s} > y] ≤ ℙ [sup_{s≤t} Z_{s} > exp(θy − 1/2θ^{2}K_{t})] ,
≤ exp(−θy + 1/2θ^{2}K_{t}).

Optimizing over θ now gives the desired result. For the second part, we establish the following bound

𝔼 ( sup_{0≤s≤t} Z_{s}) ≤ 𝔼 (exp [ sup_{0≤s≤t} Z_{s}]) ,
≤ ∫_{0}^{∞} ℙ [sup 0 ≤ s ≤ tZ_{s} ≥ log λ] dλ,
≤ 1 + ∫_{1}^{∞} exp (−(log λ)^{2}/2K_{t}) dλ < ∞.

(∗)

We have previously noted that Z is a local martingale; let T_{n} be a reducing sequence for Z, hence Z^{T_{n}} is a martingale, hence

𝔼 [Z_{t∧T_{n}}|F_{s}] = Z_{s∧T_{n}}.  (∗∗)

We note that Z_{t} is dominated by exp(θ sup_{0≤s≤t} Z_{s}), and thus by our bound we can apply the dominated convergence theorem to (∗∗) as n → ∞ to establish that Z is a true martingale.

Corollary 11.8. For all ϵ, δ > 0,

ℙ [sup_{t≥0} M_{t} ≥ ϵ & ⟨M⟩_{∞} ≤ δ] ≤ exp(−ϵ^{2}/2δ).

Proof Set T = inf{t ≥ 0 : M_{t} ≥ ϵ}, the conditions of the previous theorem now apply to M^{T}, with K_{t} = ϵ.

From this corollary, it is clear that if H is any bounded previsible process, then

exp (∫_{0}^{t} H_{s}dB_{s} − \frac{1}{2} ∫_{0}^{t} |H_{s}|^{2}ds)

is a true martingale, since this is the exponential semimartingale associated with the process ∫ HdB. Corollary 11.9. If the bounds K_{t} on ⟨M⟩ are uniform, that is if K_{t} ≤ C for all t, then the exponential martingale is Uniformly Integrable. We shall use the useful result

∫_{0}^{∞} ℙ(X ≥ log λ)dλ = ∫_{0}^{∞} 𝔼 (1_{e^{X}≥λ}) dλ = 𝔼 ∫_{0}^{∞} 1_{e^{X}≥λ}dλ = 𝔼(e^{X}).

Proof Note that the bound (∗) extends to a uniform bound

𝔼 (sup_{t≥0} Z_{t}) ≤ 1 + ∫_{1}^{∞} exp (−(log λ)^{2}/2C) dλ < ∞.

Hence Z is bounded in L^{∞} and thus a uniformly integrable martingale.

12. Lévy Characterisation of Brownian Motion

A very useful result can be proved using the Itô calculus about the characterisation of Brownian Motion. Theorem 12.1. Let {B^{i}}_{t≥0} be continuous local martingales starting from zero for i = 1, …, n. Then B_{t} = (B_{t}^{1}, …, B_{t}^{n}) is a Brownian motion with respect to (Ω, F, ℙ) adapted to the filtration F_{t}, if and only iff

⟨B^{i}, B^{j}⟩_{t} = δ_{ij}t ∀i, j ∈ {1, …, n}.

Proof In these circumstances it follows that the statement B_{t} is a Brownian Motion is by definition equivalent to stating that B_{t} − B_{s} is independent of F_{s} and is distributed normally with mean zero and covariance matrix (t − s)I.

Clearly if B_{t} is a Brownian motion then the covariation result follows trivially from the definitions. Now to establish the converse, we assume ⟨B^{i}, B^{j}⟩_{t} = δ_{ij}t for i, j ∈ {1, …, n}, and shall prove B_{t} is a Brownian Motion.

Observe that for fixed θ ∈ ℝ^{n} we can define M_{t}^{θ} by

M_{t}^{θ} := f(B_{t}, t) = exp (i(θ, x) + \frac{1}{2} |θ|^{2} t) .

By application of Itô’s formula to f we obtain (in differential form using the Einstein summation convention)

d (f(B_{t}, t)) =\frac{∂f}{∂x^{j}} (B_{t}, t)dB_{t}^{j} + \frac{∂f}{∂t} (B_{t}, t)dt + \frac{1}{2} \frac{∂^{2}f}{∂x^{j}∂x^{k}} (B_{t}, t)d⟨B^{j}, B^{k}⟩_{t},
=iθ_{j}f(B_{t}, t)dB_{t}^{j} + \frac{1}{2} |θ|^{2} f(B_{t}, t)dt − \frac{1}{2}θ_{j}θ_{k}δ_{jk}f(B_{t}, t)dt
=iθ_{j}f(B_{t}, t)dB_{t}^{j}.

Hence

M_{t}^{θ} = 1 + ∫_{0}^{t} d(f(B_{t}, t)),

and is a sum of stochastic integrals with respect to continuous local martingales and is hence itself a continuous local martingale. But note that for each t,

|M_{t}^{θ}| = (e^{\frac{1}{2} |θ|^{2}t}) < ∞

Hence for any fixed time t_{0}, (M^{t_{0}})_{t} satisfies

|(M^{t_{0}})_{t}| ≤ |(M^{t_{0}})_{∞}| < ∞,

and so is a bounded local martingale; hence (M^{t_{0}})_{t}) is a martingale. Hence M^{t_{0}} is a genuine martingale. Thus for 0 ≤ s < t we have

𝔼 (exp (i(θ, B_{t} − B_{s})) |F_{s}) = exp (−\frac{1}{2}(t − s) |θ|^{2}) a.s.

However this is just the characteristic function of a normal random variable following N(0, t − s); so by the Lévy character theorem B_{t} − B_{s} is a N(0, t − s) random variable.

13. Time Change of Brownian Motion

This result is one of frequent application, essentially it tells us that any continuous local martingale starting from zero, can be written as a time change of Brownian motion. So modulo a time change a Brownian motion is the most general kind of continuous local martingale. Proposition 13.1. Let M be a continuous local martingale starting from zero, such that ⟨M⟩_{t} → ∞ as t → ∞. Then define

τ_{s} := inf{t > 0 : ⟨M⟩_{t} > s}.

Then define

Ã_{s} := M_{τ_{s}}.

(i) This τ_{s} is an F stopping time. (ii) ⟨M⟩_{τ_{s}} = s. (iii) The local martingale M can be written as a time change of Brownian Motion as M_{t} = B_{⟨M⟩_{t}}. Moreover the process Ã_{s} is an F̃_{s} adapted Brownian Motion, where F̃_{s} is

the time-changed σ algebra i.e. F̃_{s} = F_{τ_{s}}. Proof We may assume that the map t ↦ ⟨M⟩_{t} is strictly increasing. Note that the map s ↦ τ_{s} is the inverse to t ↦ ⟨M⟩_{t}. Hence the results (i),(ii) and (iii).

Define

\begin{matrix}T_{n} := inf\{t : |M|_{t} > n\}, \\ [U_{n} := ⟨M⟩_{T_{n}}. \end{matrix}

Note that from these definitions

τ_{t∧U_{n}} = inf{s > 0 : ⟨M⟩_{s} > t ∧ U_{n}}
= inf{s > 0 : ⟨M⟩_{s} > t ∧ ⟨M⟩_{T_{n}}}
=T_{n} ∧ τ_{t}

So

Ã_{s}^{U_{n}} = Ã_{s∧U_{n}} = M_{τ_{t}}^{T_{n}}.

Now note that U_{n} is an F̃_{t} stopping time, since consider

Λ ≡ {U_{n} ≤ t} ≡ {⟨M⟩_{T_{n}} ≤ t} ≡ {T_{n} ≤ τ_{t}},

the latter event is clearly F_{τ_{t}} measurable i.e. it is F̃_{t} measurable, so U_{n} is a F̃_{t} stopping time. We may now apply the optional stopping theorem to the UI martingale M^{T_{n}}, which yields

\begin{matrix}𝔼 (Ã_{t}^{U_{n}}|F_{s}) = & 𝔼 (Ã_{t∧U_{n}}|F̃_{s}) = 𝔼 (M_{τ_{t}}^{T_{n}}|F̃_{s}) \\ = & 𝔼 (M_{τ_{t}}^{T_{n}}|F_{τ_{s}}) = M_{τ_{s}}^{T_{n}} = Ã_{s}^{U_{n}}. \end{matrix}

So Ã_{t} is a F̃_{t} local martingale. But we also know that (M^{2} − ⟨M⟩)^{T_{n}} is a UI martingale, since M^{T_{n}} is a UI martingale. By the optional stopping theorem, for 0 < r < s we have

\begin{matrix}𝔼 (Ã_{s∧U_{n}}^{2} − (s ∧ U_{n})|F̃_{r}) & = 𝔼 (((M_{τ_{s}}^{T_{n}})^{2} − ⟨M⟩_{τ_{s}∧T_{n}}) |F_{τ_{r}}) \\ = & 𝔼 ((M_{τ_{s}}^{2} − ⟨M⟩_{τ_{s}})^{T_{n}} |F_{τ_{r}}) = (M_{τ_{r}}^{2} − ⟨M⟩_{τ_{r}})^{T_{n}} \\ = & Ã_{r∧U_{n}}^{2} − (r ∧ U_{n}). \end{matrix}

Hence Ã^{2} −t is a F̃_{t} local martingale. Before we can apply Lévy’s characterisation theorem we must check that Ã is continuous; that is we must check that for almost every ω that M is constant on each interval of constancy of ⟨M⟩. By localisation it suffices to consider M a square integrable martingale, now let q be a positive rational, and define

S_{q} := inf{t > q : ⟨M⟩_{t} > ⟨M⟩_{q}},

then it is enough to show that M is constant on [q, S_{q}). But M^{2} − ⟨M⟩ is a martingale, hence

\begin{matrix}𝔼 [(M_{S_{q}}^{2} − ⟨M⟩_{S_{q}})^{2} |F_{q}] = & M_{q}^{2} − ⟨M⟩_{q} \\ = & M_{q}^{2} − ⟨M⟩_{S_{q}}, as ⟨M⟩_{q} = ⟨M⟩_{S_{q}}. \end{matrix}

Hence

𝔼 [(M_{S_{q}} − M_{q})^{2} |F_{q}] = 0,

which establishes that Ã is continuous.

Thus Ã is a continuous F̃_{t} adapted martingale with ⟨Ã_{s}⟩_{s} = s and so by the Lévy characterisation theorem Ã_{s} is a Brownian Motion.

13.1. Gaussian Martingales

The time change of Brownian Motion can be used to prove the following useful theorem. Theorem 13.2. If M is a continuous local martingale starting from zero, and ⟨M⟩_{t} is deterministic, that is if we can find a deterministic function f taking values in the non-negative real numbers such that ⟨M⟩_{t} = f(t) a.e., then M is a Gaussian Martingale (i.e. M_{t} has a Gaussian distribution for almost all t). Proof Note that by the time change of Brownian Motion theorem, we can write M_{t} as a time change of Brownian Motion through

M_{t} = B_{⟨M⟩_{t}},

where B is a standard Brownian Motion. By hypothesis ⟨M⟩_{t} = f(t), a deterministic function for almost all t, hence for almost all t,

M_{t} = B_{f(t)},

but the right hand side is a Gaussian random variable following N(0, f(t)). Hence M is a Gaussian Martingale, and at time t it has distribution given by N(0, ⟨M⟩_{t}).

As a corollary consider the stochastic integral of a purely deterministic function with respect to a Brownian motion. Corollary 13.3. Let g(t) be a deterministic function of t, then M defined by

M_{t} := ∫_{0}^{t} f(s)dB_{s},

satisfies

M_{t} ∼ N (0, ∫_{0}^{t} |f(s)|^{2}ds) .

Proof From the definition of M via a stochastic integral with respect to a continuous martingale, it is clear that M is a continuous local martingale, and by the Kunita-Watanabe result, the quadratic variation of M is given by

⟨M⟩_{t} = ∫_{0}^{t} |f(s)|ds,

hence the result follows.

This result can also be established directly in a fashion which is very similar to the proof of the Lévy characterisation theorem. Consider Z defined via

Z_{t} = exp (iθM_{t} + \frac{1}{2}θ^{2}⟨M⟩_{t}) ,

as in the Lévy characterisation proof, we see that this is a continuous local martingale, and by boundedness furthermore is a martingale, and hence

𝔼(Z_{0}) = 𝔼(Z_{t}),

whence

𝔼 (exp(iθM_{t})) = 𝔼 (exp (−\frac{1}{2}θ^{2} ∫_{0}^{t} f(s)^{2}ds))

which is the characteristic function of the appropriate normal distribution.

14. Girsanov’s Theorem

Girsanov’s theorem is an element of stochastic calculus which does not have an analogue in standard calculus.

14.1. Change of measure

When we wish to compare two measures ℙ and ℚ, we don’t want either of them simply to throw information away; since when they are positive they can be related by the Radon-Nikodym derivative; this motivates the following definition of equivalence of two measures. Definition 14.1. Two measures ℙ and ℚ are said to be equivalent if they operate on the same sample space, and if A is any event in the sample space then

ℙ(A) > 0 ⇔ ℚ(A) > 0.

In other words ℙ is absolutely continuous with respect to ℚ and ℚ is absolutely continuous with respect to ℙ. Theorem 14.2. If ℙ and ℚ are equivalent measures, and X_{t} is an F_{t}-adapted process then the following results hold

𝔼_{ℚ}(X_{t}) = 𝔼_{ℙ} (\frac{dℚ}{dℙ}X_{t}) ,

𝔼_{ℚ}(X_{t}|F_{s}) = L_{s}^{−1}𝔼_{ℙ} (L_{t}X_{t}|F_{s}) ,

where

L_{s} = 𝔼_{ℙ} ( \frac{dℚ}{dℙ}| F_{s}) .

Here L_{t} is the Radon-Nikodym derivative of ℚ with respect to ℙ. The first result basically shows that this is a martingale, and the second is a continuous time version of Bayes theorem. Proof The first part is basically the statement that the Radon-Nikodym derivative is a martingale. This follows because the measures ℙ and ℚ are equivalent, but this will not be proved in detail here. Let Y be an F_{t} measurable random variable, such that 𝔼_{ℚ}(|Y |) < ∞. We shall prove that

𝔼_{Q}(Y |F_{s}) = \frac{1}{L_{s}} 𝔼_{ℙ} [Y L_{t}|F_{s}] a.s. (ℙ and ℚ).

Then for any A ∈ F_{s}, using the definition of conditional expectation we have that

\begin{matrix}𝔼_{ℚ} (1_{A} \frac{1}{L_{s}} 𝔼_{ℙ} [Y L_{t}|F_{s}]) = & 𝔼_{ℙ} (1_{A}𝔼_{ℙ} [Y L_{t}|F_{s}]) \\ = & 𝔼_{ℙ} [1_{A}Y L_{t}] = 𝔼_{Q} [1_{A}Y] . \end{matrix}

Substituting Y = X_{t} gives the desired result.

Theorem (Girsanov). Let M be a continuous local martingale, and let Z be the associated exponential martingale

Z_{t} = exp (M_{t} − \frac{1}{2}⟨M⟩_{t}) .

If Z is uniformly integrable, then a new measure ℚ, equivalent to ℙ may be defined by

\frac{dℚ}{dℙ} = Z_{∞}.

Then if X is a continuous ℙ local martingale, X − ⟨X, M⟩ is a ℚ local martingale. Proof Since Z_{∞} exists a.s. it defines a uniformly integrable martingale (the exponential martingale), a version of which is given by Z_{t} = 𝔼(Z_{∞}|F_{t}). Hence ℚ constructed thus is a probability measure which is equivalent to ℙ. Now consider X, a ℙ local martingale. Define a sequence of stopping times which tend to infinity via

T_{n} := inf{t ≥ 0 : |X_{t}| ≥ n, or |⟨X, M⟩_{t}| ≥ n}.

Now consider the process Y defined via

Y := X^{T_{n}} − ⟨X^{T_{n}}, M⟩.

By Itô’s formula for 0 ≤ t ≤ T_{n}, remembering that dZ_{t} = Z_{t}dM_{t} as Z is the exponential martingale associated with M,

d(Z_{t}Y_{t}) =Z_{t}dY_{t} + Y_{t}dZ_{t} + d⟨Z, Y ⟩_{t}
=Z_{t}(dX_{t} − d⟨X, M⟩_{t}) + Y_{t}Z_{t}dM_{t} + d⟨Z, Y ⟩_{t}
=Z_{t}(dX_{t} − d⟨X, M⟩_{t}) + (X_{t} − ⟨X, M⟩_{t})Z_{t}dM_{t} + Z_{t}d⟨X, M⟩_{t}
=(X_{t} − ⟨X, M⟩_{t})Z_{t}dM_{t} + Z_{t}dX_{t}

Where the result d⟨Z, Y ⟩_{t} = Z_{t}d⟨X, M⟩_{t} follows from the Kunita-Watanabe theorem. Hence ZY is a ℙ-local martingale. But since Z is uniformly integrable, and Y is bounded (by construction of the stopping time T_{n}), hence ZY is a genuine ℙ-martingale. Hence for s < t and A ∈ F_{s}, we have

𝔼_{ℚ} [(Y_{t} − Y_{s})1_{A}] = 𝔼 [Z_{∞}(Y_{t} − Y_{s})1_{A}] = 𝔼 [(Z_{t}Y_{t} − Z_{s}Y_{s})1_{A}] = 0,

hence Y is a ℚ martingale. Thus X −⟨X, M⟩ is a ℚ local martingale, since T_{n} is a reducing sequence such that (X − ⟨X, M⟩)^{T_{n}} is a ℚ-martingale, and T_{n} ↑ ∞ as n → ∞.

Corollary 14.3. Let W_{t} be a ℙ Brownian motion, then W̃_{t} := W_{t} − ⟨W, M⟩_{t} is a ℚ Brownian motion. Proof Use Lévy’s characterisation of Brownian motion to see that since W̃_{t} is continuous and ⟨W̃ , W̃ ⟩_{t} = ⟨W, W ⟩_{t} = t, since W_{t} is a ℙ Brownian motion, then W̃ is a ℚ Brownian motion.

15. Brownian Martingale Representation Theorem

The following theorem has many applications, for example in the rigorous study of mathematical finance, even though the result is purely an existence theorem. The Malliavin calculus offers methods by which the process H in the following theorem can be stated explicitly, but these methods are beyond the scope of these notes! Theorem 15.1. Let B_{t} be a Brownian Motion on ℝ^{n} and G_{t} is the usual augmentation of the filtration generated by B_{t}. If Y is L^{2} integrable and is measurable with respect to G_{∞} then there exists a previsible G_{t} measurable process H_{s} uniquely defined up to evanescence such that

𝔼(Y |G_{t}) = 𝔼(Y) + ∫_{0}^{t} H_{s} · dB_{s}  (1)

The proof of this result can seem hard if you are not familiar with functional analysis style arguments. The outline of the proof is to describe all Y s which are G_{T} measurable which cannot be represented in the form (1) as belonging to the orthogonal complement of a space. Then we show that for Z in this orthogonal complement that 𝔼(ZX) = 0 for all X in a large space of G_{T} measurable functions. Finally we show that this space is sufficiently big that actually we have proved this for all G_{T} measurable functions, which includes Z so 𝔼(Z^{2}) = 0 and hence Z = 0 a.s. and we are done! Proof Without loss of generality prove the result in the case 𝔼Y = 0 where Y is L^{2} integrable and measurable with respect to G_{T} for some constant T > 0.

Define the space

L_{T}^{2} (B) = {H : H is G_{t} previsible and 𝔼 (∫_{0}^{T} ∥H_{s}∥^{2}ds) < ∞}

Consider the stochastic integral map

I : L_{T}^{2} (B) → L^{2}(G_{T})

defined by

I(H) = ∫_{0}^{T} H_{s} · dB_{s}.

As a consequence of the Itô isometry theorem, this map is an isometry. Hence the image V under I of the Hilbert space L_{T}^{2} (B) is complete and hence a closed subspace of L_{0}^{2}(G_{T}) = {H ∈ L^{2}(G_{T}) : 𝔼H = 0}. The theorem will be proved if we can establish that the image is the whole space.

We follow the usual approach in such proofs; consider the orthogonal complement of V in L_{0}^{2}(G_{T}) and we aim to show that every element of this orthogonal complement is zero. Suppose that Z is in the orthogonal complement of L_{0}^{2}(G_{T}), thus

𝔼(ZX) = 0 for all X ∈ L_{0}^{2}(G_{T})  (2)

We can define Z_{t} = 𝔼(Z|G_{t}) which is an L^{2} bounded martingale. We know that the sigma field G_{0} is trivial by the 0-1 law therefore

Z_{0} = 𝔼(Z|G_{0}) = 𝔼Z = 0.

Let H ∈ L^{2}(B) let N_{T} = I(H); we may define N_{t} = 𝔼(N_{T} |G_{t}) for 0 ≤ t ≤ T. Clearly N_{T} ∈ V as it is the image under I of some H.

Let S be a stopping time such that S ≤ T then

N_{S} = 𝔼(N_{T} |G_{S}) = 𝔼 (∫_{0}^{S} H_{s} · dB_{s} + ∫_{S}^{T} H_{s} · dB_{s}| G_{S}) = I(H1_{(0,S]}),

so consequently N_{S} ∈ V. The orthogonality relation (2) then implies that 𝔼(ZN_{S}) = 0. Thus using the martingale property of Z,

𝔼(ZN_{S}) = 𝔼(𝔼(ZN_{S}|G_{S})) = 𝔼(N_{S}𝔼(Z|G_{S})) = 𝔼(Z_{S}N_{S}) = 0

Since Z_{T} and N_{T} are square integrable, it follows that Z_{t}N_{t} is a uniformly integrable martingale.

Since the stochastic exponential of a process may be written as

M_{t} = E(iθθθ · B_{t}) = exp (iθθθ · B_{t} + \frac{1}{2}|θθθ|^{2}t) = ∫_{0}^{t} iM_{t}θθθ · dB_{t},

such a process can be taken as H = iθθθM_{t} in the definition of N_{T} and by the foregoing argument we see that Z_{t}M_{t} is a martingale. Thus

Z_{s}M_{s} = 𝔼(Z_{t}M_{t}|G_{s}) = 𝔼 (Z_{t} exp (iθθθ · B_{t} + \frac{1}{2}|θθθ|^{2}t)| G_{s})

Thus

Z_{s} exp (−\frac{1}{2}|θθθ|^{2}(t − s)) = 𝔼 (Z_{t} exp (iθθθ · (B_{t} − B_{s})| G_{s})) .

Consider a partition 0 < t_{1} < t_{2} < ⋯ < t_{m} ≤ T, and by repeating the above argument, conditioning on each G_{t_{j}} we establish that

𝔼 (Z_{T} exp (i ∑_{j} θθθ_{j} · (B_{t_{j}} − B_{t_{j−1}}))) = 𝔼 (Z_{0} exp (−\frac{1}{2} ∑(t_{j} − t_{j−1})|θθθ_{j}|^{2})) = 0,

(3)

where the last equality follows since Z_{0} = 0.

This is true for any choices of θθθ_{j} ∈ ℝ^{n} for j = 1, … n. The complex valued functions defined on (ℝ^{n})^{m} by

P^{(r)}(x_{1}, …, x_{m}) = ∑_{k=1}^{K^{(n)}} c_{k}^{(r)} exp (i ∑_{j=1}^{m} a_{j,k}^{(r)} · x_{j})

clearly separate points (i.e. for given distinct points we can choose coefficients such that the functions have distinct values at these points), form a linear space and are closed under complex conjugation. Therefore by the Stone-Weierstass theorem (see [Bollobas, 1990]), their uniform closure is the space of complex valued functions (recall that the complex variable form of this theorem only requires local compactness of the domain).

Therefore we can approximate any continuous bounded complex valued function f : (ℝ^{n})^{m} → ℂ by a sequence of such P s. But we have already shown in (3) that

𝔼 (Z_{T} P^{(r)}(B_{t_{1}}, …, B_{t_{n}})) = 0

Hence by uniform approximation we can extend this to any f continuous, bounded

𝔼 (Z_{T} f(B_{t_{1}}, …, B_{t_{n}})) = 0.

Now we use the monotone class framework; consider the class H such that for H ∈ H,

𝔼(Z_{T} H) = 0

This {calH} is a vector space, and contains the constant one since 𝔼(Z) = 0. The foregoing argument shows that it contains all H measurable with respect to the sigma field σ(B_{t_{1}}, …, B_{t_{n}}) with 0 < t_{1} < t_{2} < ⋯ < t_{n} ≤ T. Thus the monotone class theorem implies that it contains all functions which are measurable with respect to G_{T}.

The function Z_{T} ∈ G_{T}, and we have shown 𝔼(Z_{T} X) = 0 for X ∈ G_{T}. Thus we can take X = Z_{T} whence 𝔼(Z_{T}^{2} ) = 0 which implies that Z_{T} = 0 a.s.. This establishes the desired result.

The reader should examine the latter part of the proof carefully; it is in fact related to the proof that the set

{exp (i ∫_{0}^{t} θθθ_{s} · dB_{s}) : θθθ ∈ L^{∞}([0, t], ℝ^{m})}

is total in L^{1}. A set S is said to be total if 𝔼(af) = 0 for all a ∈ S implies a = 0 a.s.. The full proof of this result will reappear in a more abstract form in the stochastic filtering section of these notes.

16. Stochastic Differential Equations

Stochastic differential equations arise naturally in various engineering problems, where the effects of random ‘noise’ perturbations to a system are being considered. For example in the problem of tracking a satelite, we know that it’s motion will obey Newton’s law to a very high degree of accuracy, so in theory we can integrate the trajectories from the initial point. However in practice there are other random effects which perturb the motion.

The variety of SDE to be considered here describes a diffusion process and has the form

dX_{t} = b(t, X_{t}) + σ(t, X_{t})dB_{t},  (∗)

where b_{i}(x, t), and σ_{ij}(t, x) for 1 ≤ i ≤ d and 1 ≤ j ≤ r are Borel measurable functions.

In practice such SDEs generally occur written in the Statonowich form, but as we have seen the Itô form has numerous calculational advantages (especially the fact that local martinagles are a closed class under the Itô integral), so it is conventional to transform the SDE to the Itô form before proceeding.

Strong Solutions A strong solution of the SDE (∗) on the given probability space (Ω, F, ℙ) with initial condition ζ is a process (X_{t})_{t≥0} which has continuous sample paths such that

(i) X_{t} is adapted to the augmented filtration generated by the Brownian motion B and initial condition ζ, which is denoted F_{t}.

(ii) ℙ(X_{0} = ζ) = 1 (iii) For every 0 ≤ t < ∞ and for each 1 ≤ i ≤ d and 1 ≤ j ≤ r, then the folllowing holds

almost surely

∫_{0}^{t} (|b_{i}(s, X_{s})| + σ_{ij}^{2}(s, X_{s})) ds < ∞,

(iv) Almost surely the following holds

X_{t} = X_{0} + ∫_{0}^{t} b(s, X_{s})ds + ∫_{0}^{t} σ(s, X_{s})dB_{s}.

Lipshitz Conditions Let ∥ · ∥ denote the usual Euclidean norm on ℝ^{d}. Recall that a function f is said to be Lipshitz if there exists a constant K such that

∥f(x) − f(y)∥ ≤ K∥x − y∥,

we shall generalise the norm concept to a (d × r) matrix σ by defining

∥σ∥^{2} = ∑_{i=1}^{d} ∑_{j=1}^{r} σ_{ij}^{2}.

The concept of Liphshitz continuity can be extended to that of local Lipshitz continuity, by requiring that for each n there exists K_{n}, such that for all x and y such that ∥x∥ ≤ n and ∥y∥ ≤ n then

∥f(x) − f(y)∥ ≤ K_{n}∥x − y∥.

Strong Uniqueness of Solutions

Theorem (Uniqueness) 16.1. Suppose that b(t, x) and σ(t, x) are locally Lipshitz continuous in the spatial variable (x). That is for every n ≥ 1 there exists a constant K_{n} > 0 such that for every t ≥ 0, ∥x∥ ≤ n and ∥y∥ ≤ n the following holds

∥b(t, x) − b(t, y)∥ + ∥σ(t, x) − σ(t, y)∥ ≤ K_{n}∥x − y∥.

Then strong uniqueness holds for the pair (b, σ), which is to say that if X and X̃ are two strong solutions of (∗) relative to B with initial condition ζ then X and X̃ are indistinguishable, that is

ℙ [X_{t} = X̃_{t}∀t : 0 ≤ t < ∞] = 1.

The proof of this result is importanty inasmuch as it illustrates the first example of a technique of bounding which recurs again and again throughout the theory of stochastic differential equations. Therefore I make no apology for spelling the proof out in excessive detail, as it is most important to understand exactly where each step comes from! Proof Suppose that X and X̃ are strong solutions of (∗), relative to the same brownian motion B and initial condition ζ on the same probability space (Ω, F, ℙ). Define a sequence of stopping times

τ_{n} = inf{t ≥ 0 : ∥X_{t}∥ ≥ n}, and τ̃_{n} = inf{t ≥ 0 : ∥Y_{t}∥ ≥ n}.

Now set S_{n} = min(τ_{n}, τ̃_{n}). Clearly S_{n} is also a stopping time, and S_{n} → ∞ a.s. (ℙ) as n → ∞. These stopping times are only needed because b and σ are being assumed merely to be locally Lipshitz. If they are assumed to be Lipshitz, as will be needed in the existence part of the proof, then this complexity may be ignored.

Hence

\begin{aligned}X_{t∧S_{n}} − X̃_{t∧S_{n}} = & ∫_{0}^{t∧S_{n}} (b(u, X_{u}) − b(u, X̃_{u})) du \\ & + ∫_{0}^{t∧S_{n}} (σ(u, X_{u}) − σ(u, X̃_{u})) dW_{u}. \end{aligned}

Now we consider evaluating 𝔼∥X_{t∧S_{n}} − X̃_{t∧S_{n}}∥^{2}, the first stage follows using the identity (a + b)^{2} ≤ 4(a^{2} + b^{2}),

\begin{matrix}𝔼∥X_{t∧S_{n}} − X̃_{t∧S_{n}}∥^{2} ≤ & 4𝔼 [∫_{0}^{t∧S_{n}} (b(u, X_{u}) − b(u, X̃_{u})) du]^{2} \\ + 4𝔼 [∫_{0}^{t∧S_{n}} (σ(u, X_{u}) − σ(u, X̃_{u})) dW_{u}]^{2} \end{matrix}

Considering the second term, we use the Itô isometry which we remember states that ∥(H · M)∥_{2} = ∥H∥_{M}, so

𝔼 [∫_{0}^{t∧S_{n}} (σ(u, X_{u}) − σ(u, X̃_{u})) dW_{u}]^{2} =𝔼 [∫_{0}^{t∧S_{n}} |σ(u, X_{u}) − σ(u, X̃_{u})|^{2}du]

The classical Hölder inequality (in the form of the Cauchy Schwartz inequality) for Lebsgue integrals which states that for p, q ∈ (1, ∞), with p^{−1} + q^{−1} = 1 the following inequality is satisfied.

∫ |f(x)g(x)|dμ(x) ≤ (∫ |f(x)|^{p}dμ(x))^{1/p} (∫ |g(x)|^{q}dμ(x))^{1/q}

This result may be applied to the other term, taking p = q = 2 which yields

𝔼 [∫_{0}^{t∧S_{n}} (b(u, X_{u}) − b(u, X̃_{u})) du]^{2} ≤ 𝔼 [∫_{0}^{t∧S_{n}} |b(u, X_{u}) − b(u, X̃_{u})| du]^{2}

≤ 𝔼 [∫_{0}^{t∧S_{n}} 1ds ∫_{0}^{t∧S_{n}} (b(u, X_{u}) − b(u, X̃_{u}))^{2} ds]

≤ 𝔼 [t ∫_{0}^{t∧S_{n}} (b(u, X_{u}) − b(u, X̃_{u}))^{2}]

Thus combining these two useful inequalities and using the nth local Lipshitz relations we have that

𝔼∥X_{t∧S_{n}} − X̃_{t∧S_{n}}∥^{2} ≤4t𝔼 [∫_{0}^{t∧S_{n}} (b(u, X_{u}) − b(u, X̃_{u}))^{2}]

+ 4𝔼 [∫_{0}^{t∧S_{n}} |σ(u, X_{u}) − σ(u, X̃_{u})|^{2}du]

≤ 4(T + 1)K_{n}^{2}𝔼 ∫_{0}^{t} (X_{u∧S_{n}} − X̃_{u∧S_{n}})^{2} du

Now by Gronwall’s lemma, which in this case has a zero term outside of the integral, we see that 𝔼∥X_{t∧S_{n}} − X̃_{t∧S_{n}}∥^{2} = 0, and hence that ℙ(X_{t∧S_{n}} = X̃_{t∧S_{n}}) = 1 for all t < ∞. That is these two processes are modifications, and thus indistinguishable. Letting n → ∞ we see that the same is true for {X_{t}}_{t≥0} and {X̃_{t}}_{t≥0}.

Now we impose Lipshitz conditions on the functions b and σ to produce an existence result. The following form omits some measure theoretic details which are very important; for a clear treatment see Chung & Williams chapter 10.

Theorem (Existence) 16.2. If the coefficients b and σ satisfy the global Lipshitz conditions that for all u, t

b(u, x) − b(u, y) ≤ K|x − y|, |σ(t, x) − σ(t, y)| ≤ K|x − y|,

and additionally the bounded growth condition

|b(t, x)|^{2} + |σ(t, x)|^{2} ≤ K^{2}(1 + |x|^{2})

Fix a probability space (Ω, F, ℙ). Let ξ be a random valued vector, independent of the Brownian Motion B_{t}, with finite second moment. Let F_{t} be the augmented filtration associated with the Brownian Motion B_{t} and ξ. Then there exists a continuous, adapted process X which is a strong solution of the SDE with initial condition ξ. Additionally this process is square integrable: for each T > 0 there exists C(K, T) such that

𝔼|X_{t}|^{2} ≤ C (1 + 𝔼|ξ|^{2}) e^{Ct},

for 0 ≤ t ≤ T. Proof This proof proceeds by Picard iteration through a map F, analogously to the deterministic case to prove the existence of solutions to first order ordinary differential equations. This is a departure from the more conventional proof of this result. Let F be a map from the space C_{T} of continuous adapted processes X from Ω × [0, T] to ℝ, such that 𝔼 [(sup_{t≤T} X_{t})^{2}] <

∞. Define X_{t}^{(k)} recursively, with X_{t}^{(0)} = ξ, and

X_{t}^{(k+1)} = F (X^{k})_{t} = ξ + ∫_{0}^{t} b(s, X_{s}^{(k)})ds + ∫_{0}^{t} σ(s, X_{s}^{(k)})dB_{s}

[Note: we have left out checking that the image of X under F is indeed adapted!] Now note that using (a + b)^{2} ≤ 2a^{2} + ab^{2}, we have using the same bounds as in the uniqueness result that

𝔼 [( sup_{0≤t≤T} F (X)_{t} − F (Y)_{t})^{2}] ≤ 2𝔼 (sup_{t≤T} |∫_{0}^{T} (σ(X_{s}) − σ(Y_{s})) dB_{s}|^{2})

+ 2𝔼 (sup_{t≤T} |∫_{0}^{T} (b(X_{s}) − b(Y_{s})) ds|^{2})

≤ 2K^{2}(4 + T) ∫_{0}^{T} 𝔼 [(sup_{t≤T} |X_{t} − Y_{t}|^{2})^{2}] dt.

By induction we see that for each T we can prove

𝔼 [(sup_{t≤T} F^{n}(X) − F^{n}(Y))^{2}] ≤ \frac{C^{n}T^{n}}{n!} 𝔼 [(sup_{t≤T} X_{t} − Y_{t})^{2}]

So by taking n sufficiently large we have that F^{n} is a contraction mapping and so by the contraction mapping theorem, F^{n} mapping C_{T} to itself has a fixed point, which must be unique, call it X^{(T)}. Clearly from the uniqueness part X_{t}^{(T)} = X_{t}^{(T′)} for t ≤ T ∧ T′ a.s., and so we may consistently define X ∈ C by

X_{t} = X_{t}^{(N)} for t ≤ N, N ∈ ℕ,

which solves the SDE, and has already been shown to be unique.

17. Relations to Second Order PDEs

The aim of this section is to show a rather surprising connection between stochastic differential equations and the solution of second order partial differential equations. Surprising though the results may seem they often provide a viable route to calculating the solutions of explicit PDEs (an example of this is solving the Black-Scholes Equation in Option Pricing, which is much easier to solve via stochastic methods, than as a second order PDE). At first this may well seem to be surprising since one problem is entirely deterministic and the other in inherently stochastic!

17.1. Infinitesimal Generator

Consider the following d-dimensional SDE,

dX_{t} =b(X_{t})dt + σ(X_{t})dB_{t},
X_{0} =x_{0}

where σ is a d × d matrix with elements σ = {σ_{ij}}. This SDE has infinitesimal generator A, where

A = ∑_{j=1}^{d} b^{k}(X_{t})\frac{∂}{∂x^{j}} + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} ∑_{k=1}^{d} σ_{ik}(X_{t})σ_{kj}(X_{t})\frac{∂^{2}}{∂x^{i}∂x^{j}} .

It is conventional to set

a_{ij} = ∑_{k=1}^{d} σ_{ik}σ_{kj},

whence A takes the simpler form

A = ∑_{j=1}^{d} b^{j}(X_{t})\frac{∂}{∂x^{j}} + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} a_{ij}(X_{t})\frac{∂^{2}}{∂x^{i}∂x^{j}} .

Why is the definition useful? Consider application of Itô’s formula to f(X_{t}), which yields

f(X_{t}) − f(X_{0}) = ∫_{0}^{t} ∑_{j=1}^{d} \frac{∂f}{∂x^{j}} (X_{s})dX_{s} + \frac{1}{2} ∫_{0}^{t} ∑_{i=1}^{d} ∑_{j=1}^{d} \frac{∂^{2}f}{∂x^{i}∂x^{j}} (X_{s})d⟨X^{i}, X^{j}⟩_{s}.

Substituting for dX_{t} from the SDE we obtain,

f(X_{t}) − f(X_{0}) = ∫_{0}^{t} (∑_{j=1}^{d} b^{j}(X_{s})\frac{∂f}{∂x^{j}} (X_{s}) + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} ∑_{k=1}^{d} σ_{ik}σ_{kj} \frac{∂^{2}f}{∂x^{i}∂x^{j}} (X_{s})) dt

+ ∫_{0}^{t} ∑_{j=1}^{d} σ_{ij}(X_{s})\frac{∂f}{∂x^{j}} (X_{s})dB_{s}

= ∫_{0}^{t} Af(X_{s})ds + ∫_{0}^{t} ∑_{j=1}^{d} σ_{ij}(X_{s})\frac{∂f}{∂x^{j}} (X_{s})dB_{s}

Definition 17.1. We say that X_{t} satisfies the martingale problem for A, if X_{t} is F_{t} adapted and

M_{t} = f(X_{t}) − f(X_{0}) − ∫_{0}^{t} Af(X_{s})ds,

is a martingale for each f ∈ C_{c}^{2}(ℝ^{d}).

It is simple to verify from the foregoing that any solution of the associated SDE solves the martingale problem for A. This can be generalised if we consider test functions ϕ ∈ C^{2}(ℝ^{+} × ℝ^{d}, ℝ), and define

M_{t}^{ϕ} := ϕ(t, X_{t}) − ϕ(0, X_{0}) − ∫_{0}^{t} (\frac{∂}{∂s} + A) ϕ(s, X_{s})ds.

then M_{t}^{ϕ} is a local martingale, for X_{t} a solution of the SDE associated with the infinitesimal generator A. The proof follows by an application of Itô’s formula to M_{t}^{ϕ}, similar to that of the above discussion.

17.2. The Dirichlet Problem

Let Ω be a subspace of ℝ^{d} with a smooth boundary ∂Ω. The Dirichlet Problem for f is defined as the solution of the system

\begin{matrix}Au + ϕ = 0 on Ω, \\ u = f on ∂Ω. \end{matrix}

Where A is a second order partial differential operator of the form

A = ∑_{j=1}^{d} b^{j}(X_{t})\frac{∂}{∂x^{j}} + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} a_{ij}(X_{t})\frac{∂^{2}}{∂x^{i}∂x^{j}} ,

which is associated as before to an SDE. This SDE will play an important role in what is to follow.

A simple example of a Dirichlet Problem is the solution of the Laplace equation in the disc, with Dirichlet boundary conditions on the boundary, i.e.

\begin{matrix}∇^{2}u = 0 on D, \\ u = f on ∂D. \end{matrix}

Theorem 17.2. For each f ∈ C_{b}^{2}(∂Ω) there exists a unique u ∈ C_{b}^{2}(\overline{Ω}) solving the Dirichlet problem for f. Moreover there exists a continuous function m : \overline{Ω} → (0, ∞) such that for all f ∈ C_{b}^{2}(∂Ω) this solution is given by

u(x) = ∫_{∂Ω} m(x, y)f(y)σ(dy).

Now remember the SDE which is associated with the infinitesimal generator A:

dX_{t} =b(X_{t})dt + σ(X_{t})dB_{t},
X_{0} =x_{0}

Often in what follows we shall want to consider the conditional expectation and probability measures, conditional on x_{0} = x, these will be denoted 𝔼_{x} and ℙ_{x} respectively. Theorem (Dirichlet Solution). Define a stopping time via

T := inf{t ≥ 0 : X_{t} ∉ Ω}.

Then u(x) given by

u(x) := 𝔼_{x} [∫_{0}^{T} ϕ(X_{s})ds + f(X_{T})] ,

solves the Dirichlet problem for f. Proof Define

M_{t} := u(X_{T ∧t}) + ∫_{0}^{t∧T} ϕ(X_{s})ds.

We shall now show that this M_{t} is a martingale. For t ≥ T, it is clear that dM_{t} = 0. For t < T by Itô’s formula

dM_{t} = du(X_{t}) + ϕ(X_{t})dt.

Also, by Itô’s formula,

du(X_{t}) = ∑_{j=1}^{d} \frac{∂u}{∂x^{i}} (X_{t})dX_{t}^{i} + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} \frac{∂^{2}u}{∂x^{i}∂x^{j}} (X_{t})d⟨X^{i}, X^{j}⟩_{t}

= ∑_{j=1}^{d} \frac{∂u}{∂x^{j}} (X_{t}) [b(X_{t})dt + σ(X_{t})dB_{t}] + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} ∑_{k=1}^{d} σ_{ik}σ_{kj} \frac{∂^{d}u}{∂x^{i}∂x^{j}} (X_{t})dt

=Au(X_{t})dt + ∑_{j=1}^{d} σ(X_{t})\frac{∂u}{∂x^{j}} (X_{t})dB_{t}.

Putting these two applications of Itô’s formula together yields

dM_{t} = (Au(X_{t}) + ϕ(X_{t})) dt + ∑_{j=1}^{d} σ(X_{t})\frac{∂u}{∂x^{j}} (X_{t})dB_{t}.

but since u solves the Dirichlet problem, then

(Au + ϕ)(X_{t}) = 0,

hence

dM_{t} = (Au(X_{t}) + ϕ(X_{t})) dt + ∑_{j=1}^{d} σ(X_{t})\frac{∂u}{∂x^{j}} (X_{t})dB_{t}^{j},

= ∑_{j=1}^{d} σ(X_{t})\frac{∂u}{∂x^{j}} (X_{t})dB_{t}^{j}.

from which we conclude by the stability property of the stochastic integral that M_{t} is a local martingale. However M_{t} is uniformly bounded on [0, t], and hence M_{t} is a martingale.

In particular, let ϕ(x) ≡ 1, and f ≡ 0, by the optional stopping theorem, since T ∧ t is a bounded stoppping time, this gives

u(x) = 𝔼_{x}(M_{0}) = 𝔼_{x}(M_{T ∧t}) = 𝔼_{x} [u(X_{T ∧t}) + (T ∧ t)].

Letting t → ∞, we have via monotone convergence that 𝔼_{x}(T) < ∞, since we know that the solutions u is bounded from the PDE solution existance theorem; hence T < ∞ a.s.. We cannot simply apply the optional stopping theorem directly, since T is not necessarily a bounded stopping time. But for arbitrary ϕ and f, we have that

|M_{t}| ≤ ∥u∥_{∞} + T ∥ϕ∥_{∞} = sup_{x∈\overline{Ω}} |u(x)| + T sup_{x∈\overline{Ω}} |ϕ(x)|,

whence as 𝔼_{x}(T) < ∞, the martingale M is uniformly integrable, and by the martingale convergence theorem has a limit M_{∞}. This limiting random variable is given by

M_{∞} = f(X_{T}) + ∫_{0}^{T} ϕ(X_{s})ds.

Hence from the identity 𝔼_{x}M_{0} = 𝔼_{x}M_{∞} we have that,

u(x) = 𝔼_{x}(M_{0}) = 𝔼_{x}(M_{∞}) = 𝔼_{x} [f(X_{T}) + ∫_{0}^{T} ϕ(X_{s})ds] .

17.3. The Cauchy Problem

The Cauchy Problem for f, a C_{b}^{2} function, is the solution of the system

\frac{∂u}{∂t} = Au on Ω

u(0, x) = f(x) on x ∈ Ω

u(t, x) = f(x) ∀t ≥ 0, on x ∈ ∂Ω

A typical problem of this sort is to solve the heat equation,

\frac{∂u}{∂t} = \frac{1}{2}∇^{2}u,

where the function u represents the temperature in a region Ω, and the boundary condition is to specify the temperature field over the region at time zero, i.e. a condition of the form

u(0, x) = f(x) for x ∈ Ω,

In addition the boundary has its temperature fixed at zero,

u(0, x) = 0 for x ∈ ∂Ω.

If Ω is just the real line, then the solution has the beautifully simple form

u(t, x) = 𝔼_{x} (f(B_{t})) ,

where B_{t} is a standard Brownian Motion. Theorem (Cauchy Existence) 17.3. For each f ∈ C_{b}^{2}(ℝ^{d}) there exists a unique u in C_{b}^{1,2}(ℝ×ℝ^{d}) such that u solves the Cauchy Problem for f. Moreover there exists a continuous function (the heat kernel)

p : (0, ∞) × ℝ^{d} × ℝ^{d} → (0, ∞),

such that for all f ∈ C_{b}^{2}(ℝ^{d}), the solution to the Cauchy Problem for f is given by

u(t, x) = ∫_{ℝ^{d}} p(t, x, y)f(y)dy.

Theorem 17.4. Let u ∈ C_{b}^{1,2}(ℝ × ℝ^{d}) be the solution of the Cauchy Problem for f. Then define

T := inf{t ≥ 0 : X_{t} ∉ Ω},

a stopping time. Then

u(t, x) = 𝔼_{x} [f(X_{T ∧t})]

Proof Fix s ∈ (0, ∞) and consider the time reversed process

M_{t} := u((s − t) ∧ T, X_{t∧T}).

There are three cases now to consider; for 0 ≤ T ≤ t ≤ s, M_{t} = u((s − t) ∧ T, X_{T}), where X_{T} ∈ ∂Ω, so from the boundary condition, M_{t} = f(X_{T}), and hence it is clear that dM_{t} = 0. For 0 ≤ s ≤ T ≤ t and for 0 ≤ t ≤ s ≤ T, the argument is similar; in the latter case by Itô’s formula we obtain

dM_{t} = − \frac{∂u}{∂t}(s − t, X_{t})dt + ∑_{j=1}^{d} \frac{∂u}{∂x^{j}} (s − t, X_{t})dX_{t}^{j}

+ \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} \frac{∂^{2}u}{∂x^{i}∂x^{j}} (s − t, X_{t})d⟨X^{i}, X^{j}⟩_{t},

= (−\frac{∂u}{∂t} + Au) (s − t, X_{t})dt + ∑_{j=1}^{d} \frac{∂u}{∂x^{j}} (s − t, X_{t}) ∑_{k=1}^{d} σ_{jk}(X_{t})dB_{t}^{k}.

We obtain a similar result in the 0 ≤ t ≤ T ≤ s, case but with s replaced by T. Thus for u solving the Cauchy Problem for f, we have that

(−\frac{∂u}{∂t} + Au) = 0,

we see that M_{t} is a local martingale. Boundedness implies that M_{t} is a martingale, and hence by optional stopping

u(s, x) = 𝔼_{x}(M_{0}) = 𝔼_{x}(M_{s}) = 𝔼_{x}(f(X_{s∧T})),

17.4. Feynman-Kac̆ Representation

Feynman observed the following representation for the representation of the solution of a PDE via the expectation of a suitable function of a Brownian Motion ‘intuitively’ and the theory was later made rigorous by Kac̆.

In what context was Feynman interested in this problem? Consider the Schrödinger Equation,

−\frac{ℏ^{2}}{2m}∇^{2}Φ(x, t) + V (x)Φ(x, t) = iℏ\frac{∂}{∂t}Φ(x, t),

which is a second order PDE. Feynman introduced the concept of a path-integral to express solutions to such an equation. In a manner which is analogous to the ‘Hamiltonian’ principle in classical mechanics, there is an action integral which is minimised over all ‘permissible paths’ that the system can take.

We have already considered solving the Cauchy problem

\frac{∂u}{∂t} = Au on Ω

u(0, x) = f(x) on x ∈ Ω

u(t, x) = f(x) ∀t ≥ 0, on x ∈ ∂Ω

where A is the generator of an SDE and hence of the form

A = ∑_{j=1}^{d} b^{j}(X_{t})\frac{∂}{∂x^{j}} + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} a_{ij}(X_{t})\frac{∂^{2}}{∂x^{i}∂x^{j}} .

Now consider the more general form of the same Cauchy problem where we consider a Cauchy Problem with generator L of the form:

L ≡ A + v = ∑_{j=1}^{d} b^{j}(X_{t})\frac{∂}{∂x^{j}} + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} a_{ij}(X_{t})\frac{∂^{2}}{∂x^{i}∂x^{j}} + v(X_{t}).

For example let X_{t} = B_{t} corresponding to

A = \frac{1}{2}∇^{2}, L = \frac{1}{2}∇^{2} + v(X_{t}),

so in this example we are solving the problem

\frac{∂u}{∂t} = \frac{1}{2}∇^{2}u(t, X_{t}) + v(X_{t})u(X_{t}) on ℝ^{d}.

u(0, x) = f(x) on ∂ℝ^{d}.

The Feynman-Kac̆ Representation Theorem expresses the solution of a general second order PDE in terms of an expectation of a function of a Brownian Motion. To simplify the statement of the result, we shall work on Ω = ℝ^{d}, since this removes the problem of considering the Brownian Motion hitting the boundary. Theorem (Feynman-Kac̆ Representation). Let u ∈ C_{b}^{1,2}(ℝ × ℝ^{d}) be a solution of the Cauchy Problem with a generator of the above form with initial condition f, and let X_{t} be a solution of the SDE with infinitessimal generator A in ℝ^{d} starting at x. Then

u(t, x) = 𝔼_{x} [f(X_{t}) exp (∫_{0}^{t} v(X_{s})ds)] .

Proof Note that the SDE for X_{t} has the form

dX_{t} = a(X_{t})dB_{t} + b(X_{t})dt.

Relations to Second Order PDEs 68

Fix s ∈ (0, ∞) and apply Itô’s formula to

M_{t} = u(s − t, X_{t}) exp (∫_{0}^{t} v(X_{r})dr) .

For notational convenience, let

E_{t} = exp (∫_{0}^{t} v(X_{r})dr) ,

whence by Itô’s formula dE_{t} = E_{t}v(X_{t})dt. For 0 ≤ t ≤ s, we have

\begin{matrix}dM_{t} = & ∑_{j=1}^{d} \frac{∂u}{∂x^{j}} (s − t, X_{t})E_{t}dB_{t}^{j} + (−\frac{∂u}{∂t} + Au + vu) (s − t, X_{t})E_{t}dt \\ = & ∑_{j=1}^{d} \frac{∂u}{∂x^{j}} (s − t, X_{t})E_{t}dB_{t}^{j}. \end{matrix}

Hence M_{t} is a local martingale; since it it bounded, M_{t} is a martingale and hence by optional stopping

u(s, x) = 𝔼_{x}(M_{0}) = 𝔼_{x}(M_{s}) = 𝔼_{x} (f(X_{s})E_{s}) .

18. Stochastic Filtering

Engineers have for many years studied system problems arising in the design of control systems to be placed between user inputs to control system parameters in order to achieve a desired response. For example the connection between the flight controls on a modern aircraft such as the Harrier Jump Jet and the aircraft control surfaces go via a complex control system rather than simple wire cables as found in early aircraft.

Even a basic control system such as a central heating thermostat is based upon an element of feedback – observing the state of the system (room temperature) and adjusting the system inputs (turning the boiler off or on) to keep the system state close to the desired value (the temperature set on the dial).

In the example of a high performance aircraft, sudden movement of an aircraft control surface communicated directly to the control surfaces might cause uncontrolled growing oscillations – which was not the response required and the control system would have to modify the control surfaces to avoid this happening.

In the real world the behaviour of these systems is affected by extra random unknown factors which can not be known; for example wind effects on the airframe, engine vibration. Also all measurements have inherent random errors which cannot be eliminated completely. Attempts to solve control problems in this context require that we have some idea how to use the observations made to give a best possible estimate of the system state at a given time.

In this basic introduction to stochastic non-linear filtering we shall consider a system described by a stochastic process called the signal process whose behaviour is governed by a dynamical system containing a Brownian motion noise input. This is observed, together with unavoidable observation noise. The goal of the stochastic filtering problem is to determine the law of the signal given the observations. This must be formalised in the language of stochastic calculus. The notation ∥ · ∥ will be used in this section to denote the Euclidean norm in d-dimensional space.

An important modern example of stochastic filtering is the use of GPS measurements to correct an inertial navigation system (INS). An INS system has transducers which measure acceleration. Starting from a known point in space the acceleration can be integrated twice to compute a position. But since there are error in acceleration measurements this position will gradually diverge from the true position. This is a phenomenon familiar to anyone who has navigated with a map by dead-reckoning (From the station I walk NE for a mile, E for 2 miles – where am I?).

The simple navigator checks his position as frequently as possible by other means (e.g. observing the bearing to several landmarks, taking a star-sight or by looking at a handheld GPS receiver).

In a similar fashion we might try and correct our INS position estimates by using periodic GPS observations; these GPS position measurements can not be made continuously and are subject to random errors; additionally the observer continues to move while taking the readings. What is the optimal way to use these GPS measurements to correct our INS estimate? The answer lies in the solution of a stochastic filtering problem. The signal process is the true position and the observation process is the GPS position process.

18.1. Signal Process

Let {X_{t}, F_{t}, t ≥ 0} be the signal process. It is defined to be the solution of the stochastic differential equation

dX_{t} = f(t, X_{t})dt + σ(t, X_{t})dV_{t},  (4)

where V is a d dimensional standard Brownian motion. The coefficients satisfy the conditions

∥f(t, x) − f(t, y)∥ + ∥σ(t, x) − σ(t, y)∥ ≤ k∥x − y∥
∥f(t, x)∥^{2} + ∥σ(t, x)∥^{2} ≤ k^{2}(1 + ∥x∥^{2}),

which ensure that the equation for X has a unique solution.

18.2. Observation Process

The observation process satisfies the stochastic differential equation

dY_{t} = h(t, X_{t})dt + dW_{t}, Y_{0} = 0,  (5)

where W_{t} is an m dimensional standard Brownian motion independent of V_{t}. The function h taking values in ℝ^{m} satisfies a linear growth condition

∥h(t, x)∥^{2} ≤ k(1 + ∥x∥^{2}).

A consequence of this condition is that for any T > 0,

𝔼 (∥X_{t}∥^{2}) ≤ C (1 + 𝔼 [∥X_{0}∥^{2}]) e^{ct},  (6)

for t ∈ [0, T] with suitable constants C and c which may be functions of T. As a result of this bound

𝔼 [∫_{0}^{T} ∥h(s, X_{s})∥^{2} ds] < ∞.

Given this process a sequence of observation σ-algebras may be defined

Y_{t}^{o} := σ (Y_{s} : 0 ≤ s ≤ t) .

These must be augmented in the usual fashion to give

Y_{t} = Y_{t}^{o} ∪ N ,

where N is the set of ℙ null subsets of Ω, and

Y := ⋁_{t≥0} Y_{t}.

18.3. The Filtering Problem

The above has set the scene, we have a real physical system whose state at time t is represented by the vector X_{t}. The system state is governed by an SDE with a noise term representing random perturbations to the system. This is observed to give the observation process Y_{t} which includes new independent noise (represented by W_{t}).

The filtering problem is to find the conditional law of the signal process given the observations to date, i.e. to find

π_{t}(ϕ) := 𝔼 (ϕ(X_{t})|Y_{t}) ∀t ∈ [0, T].  (7)

The following proofs are in many cases made complex by the fact that we do not assume that the functions f, h and σ are bounded.

18.4. Change of Measure

To solve the filtering problem Girsanov’s theorem will be used to make a change of measure to a new measure under which the observation process Y is a Brownian motion. Let

Z_{t} := exp (− ∫_{0}^{t} h^{T} (s, X_{s})dW_{s} − \frac{1}{2} ∫_{0}^{t} ∥h(s, X_{s})∥^{2} ds) ,  (8)

be a real valued process. Proposition 18.1. The process Z_{t} is a martingale with respect to the filtration {F_{t}, t ≥ 0}. Proof The process Z_{t} satisfies

Z_{t} = 1 − ∫_{0}^{t} Z_{s}h^{T} (s, X_{s})dW_{s},

that is Z_{t} solves the SDE

\begin{aligned}dZ_{t} & = −Z_{t}h^{T} (t, X_{t})dW_{t}, \\ Z_{0} & = 1. \end{aligned}

Given this expression, Z_{t} is a positive continuous local martingale and hence is a supermartingale. To prove that it is a true martingale we must prove in addition that it has constant mean.

Let T_{n} be a reducing sequence for the local martingale Z_{t}, i.e. an increasing sequence of stopping times tending to infinity as n → ∞ such that for each n, Z^{T_{n}} is a genuine martingale. By Fatou’s lemma, and the local martingale property

𝔼Z_{t} = 𝔼 lim_{n→∞} Z_{t}^{T_{n}} ≤ lim inf 𝔼Z_{t}^{T_{n}} = 𝔼Z_{0}^{T_{n}} = 1,

so

𝔼Z_{t} ≤ 1 ∀t.  (9)

This will be used as an upper bound in an application of the dominated convergence theorem. By application of Itô’s formula to the function f defined by

f(x) = \frac{x}{1 + ϵx},

we obtain that

f(Z_{t}) = f(Z_{0}) + ∫_{0}^{t} f′(Z_{s})dZ_{s} + \frac{1}{2} ∫_{0}^{t} f′′(Z_{s})d⟨Z_{s}, Z_{s}⟩,

Hence

\frac{Z_{t}}{1 + ϵZ_{t}} = \frac{1}{1 + ϵ} − ∫_{0}^{t} \frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}}dW_{s} − ∫_{0}^{t} \frac{ϵZ_{s}^{2}∥h(s, X_{s})∥^{2}}{(1 + ϵZ_{s})^{3}} ds.  (10)

Consider the term

∫_{0}^{t} \frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}}dW_{s},

clearly this a local martingale, since it is a stochastic integral. The next step in the proof is explained in detail as it is one which occurs frequently. We wish to show that the above stochastic integral is in fact a genuine martingale. From the earlier theory (for L^{2} integrable martingales) it suffices to show that integrand is in L^{2}(W). We therefore compute

∥\frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}} ∥_{W} = 𝔼 [∫_{0}^{t} ∥\frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}} ∥^{2} ds] ,

with a view to showing that it is finite, whence the integrand must be in L^{2}(W) and hence the integral a genuine martingale. We note that since Z_{s} ≥ 0, that

\frac{1}{(1 + ϵZ_{s})} ≤ 1, and \frac{Z_{s}^{2}}{(1 + ϵZ_{s})^{2}} ≤ \frac{1}{ϵ^{2}} ,

so,

∥\frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}} ∥^{2} =\frac{Z_{s}^{2}}{(1 + ϵZ_{s})^{4}} ∥h(s, X_{s})∥^{2}
= [\frac{Z_{s}^{2}}{(1 + ϵZ_{s})^{2}} ] × [\frac{1}{(1 + ϵZ_{s})^{2}} ] × ∥h(s, X_{s})∥^{2}
≤\frac{1}{ϵ^{2}} ∥h(s, X_{s})∥^{2} .

Using this inequality we obtain

𝔼 [∫_{0}^{t} ∥\frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}} ∥^{2} ds] ≤ \frac{1}{ϵ^{2}} 𝔼 [∫_{0}^{T} ∥h(s, X_{s})∥^{2} ds] ,

however as a consequence of our linear growth condition on h, the last expectation is finite. Therefore

∫_{0}^{t} \frac{Z_{s}h^{T} (s, X_{s})}{(1 + ϵZ_{s})^{2}}dW_{s}

is a genuine martingale, and hence taking the expectation of (10) we obtain

𝔼 [\frac{Z_{t}}{1 + ϵZ_{t}} ] = \frac{1}{1 + ϵ} − 𝔼 [∫_{0}^{t} \frac{ϵZ_{s}^{2}∥h(s, X_{s})∥^{2}}{(1 + ϵZ_{s})^{3}} ds] .

Consider the integrand on the right hand side; clearly it tends to zero as ϵ → 0. But also

\begin{matrix}\frac{ϵZ_{s}^{2}∥h(s, X_{s})∥^{2}}{(1 + ϵZ_{s})^{3}} = & \frac{ϵZ_{s}}{(1 + ϵZ_{s})^{3}} · Z_{s} · ∥h(s, X_{s})∥^{2} \\ ≤ & \frac{1 + ϵZ_{s}}{(1 + ϵZ_{s})^{3}} · Z_{s} · ∥h(s, X_{s})∥^{2} \\ ≤ & kZ_{s} (1 + ∥X_{s}∥^{2}) , \end{matrix}

where we have used the fact that ∥h(s, X_{s})∥^{2} ≤ k(1 + ∥X_{s}∥^{2}). So from the fact that 𝔼Z_{s} ≤ 1 and using the next lemma we shall see that 𝔼 (Z_{s}∥X_{s}∥^{2}) ≤ C. Hence we conclude that kZ_{s}(1 + ∥X_{s}∥^{2}) is an integrable dominating function for the integrand on interest. Hence by the Dominated Convergence Theorem, as ϵ → 0,

𝔼 [∫_{0}^{t} \frac{ϵZ_{s}^{2}∥h(s, X_{s})∥^{2}}{(1 + ϵZ_{s})^{3}} ds] → 0.

In addition, since 𝔼(Z_{s}) ≤ 1, the bounded convergence theorem yields

𝔼 (\frac{Z_{s}}{1 + ϵZ_{s}} ) → 𝔼(Z_{s}), as ϵ → 0.

Hence we conclude that

𝔼(Z_{t}) = 1 ∀t,

and so Z_{t} is a genuine martingale.

Lemma 18.2.

𝔼 [∫_{0}^{t} Z_{s}∥X_{s}∥^{2}ds] < C(T) ∀t ∈ [0, T].

Proof To establish this result Itô’s formula is used to derive two important results

d(∥X_{t}∥^{2}) =2X_{t}^{T} (fdt + σdW_{s}) + tr(σ^{T} σ)dt
d(Z_{t}∥X_{t}∥^{2}) = − Z_{t}∥X_{t}∥^{2}h^{T} dV_{s} + Z_{t} (2X_{t}^{T} (fdt + σdW_{s}) + tr(σσ^{T})) dt

By Itô’s formula,

d (\frac{Z_{t}∥X_{t}∥^{2}}{1 + ϵZ_{t}∥X_{t}∥^{2}} ) = \frac{1}{(1 + ϵZ_{t}∥X_{t}∥^{2})^{2}} d (Z_{t}∥X_{t}∥^{2}) + \frac{ϵ}{(1 + ϵZ_{t}∥X_{t}∥^{2})^{3}} d⟨Z_{t}∥X_{t}∥^{2}⟩.

Substituting for d(Z_{t}∥X_{t}∥^{2}) into this expression yields

d (\frac{Z_{t}∥X_{t}∥^{2}}{1 + ϵZ_{t}∥X_{t}∥^{2}} ) = \frac{1}{(1 + ϵZ_{t}∥X_{t}∥^{2})^{2}} [−Z_{t}∥X_{t}∥^{2}h^{T} dW_{t} + Z_{t}2X_{t}^{T} σdV_{s}]

+ [\frac{Z_{t} (2X_{t}^{T} f + tr(σσ^{T}))}{(1 + ϵZ_{t}∥X_{t}∥^{2})^{2}} − \frac{ϵ (Z_{t}^{2}∥X_{t}∥^{4}h^{T} h + 4Z_{t}^{2}2X_{t}^{T} σσ^{T} X_{t})}{(1 + ϵZ_{t}∥X_{t}∥^{2})^{3}}] dt.

After integrating this expression from 0 to t, the terms which are stochastic integrals are local martingales. In fact they can be shown to be genuine martingales. For example consider the term

∫_{0}^{t} \frac{Z_{s}2X_{s}^{T} σ}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{2}} dV_{s},

to show that this is a martingale we must therefore establish that

𝔼 [∫_{0}^{t} [\frac{Z_{s}2X_{s}^{T} σ}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{2}} ]^{2} ds] = 4𝔼 [∫_{0}^{t} \frac{Z_{s}^{2}X_{s}^{T} σσ^{T} X_{s}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}} ds] < ∞.

In order to establish this inequality consider the term X_{t}^{T} σσ^{T} X_{t} which is a sum over terms of the form

|X_{t}^{i}σ_{ij}(t, X_{t})σ_{kj}(t, X_{t})X_{t}^{k}| ≤ ∥X_{t}∥^{2}∥σ∥^{2},

of which there are d^{3} terms (in ℝ^{d}). But by the linear increase condition on σ, we have for some constant κ

∥σ∥^{2} ≤ κ(1 + ∥X∥^{2}),

and hence

|X_{t}^{i}σ_{ij}(t, X_{t})σ_{kj}(t, X_{t})X_{t}^{k}| ≤ κ∥X_{t}∥^{2} (1 + ∥X_{t}∥^{2}) ,

so the integral may be bounded by

∫_{0}^{t} \frac{Z_{s}^{2}X_{s}^{T} σσ^{T} X_{s}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}} ds ≤ κd^{3} ∫_{0}^{t} \frac{Z_{s}^{2}∥X_{s}∥^{2} (1 + ∥X_{s}∥^{2})}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}}ds

= κd^{3} [∫_{0}^{t} \frac{Z_{s}^{2}∥X_{s}∥^{2}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}} ds + ∫_{0}^{t} \frac{Z_{s}^{2}∥X_{s}∥^{4}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}} ] ds

Considering each term separately, the first satisfies

∫_{0}^{t} \frac{Z_{s}^{2}∥X_{s}∥^{2}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}} ds ≤ ∫_{0}^{t} Z_{s} × \frac{Z_{s}∥X_{s}∥^{2}}{(1 + ϵZ_{s}∥X_{t}∥^{2})} × \frac{1}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{3}} ds

≤ ∫_{0}^{t} \frac{Z_{s}}{ϵ} ds ≤ \frac{1}{ϵ} ∫_{0}^{t} Z_{s}.

The last integral has a bounded expectation since 𝔼 (Z_{s}) ≤ 1. Similarly for the second term,

∫_{0}^{t} \frac{Z_{s}^{2}∥X_{s}∥^{4}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{4}} ds ≤ ∫_{0}^{t} \frac{Z_{s}^{2}∥X_{s}∥^{4}}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{2}} × \frac{1}{(1 + ϵZ_{s}∥X_{s}∥^{2})^{2}} ds ≤ \frac{1}{ϵ^{2}} t < ∞.

A similar argument holds for the other stochastic integrals, so they must both be genuine martingales, and hence if we take the expectations of the integrals they are zero. Integrating the whole expression from 0 to t, then taking the expectation and finally differentiation with respect to t yields

\frac{d}{dt}𝔼 (\frac{Z_{t}∥X_{t}∥^{2}}{1 + ϵZ_{t}∥X_{t}∥^{2}} ) ≤ 𝔼 (\frac{Z_{t}(2X_{t}^{T} f + tr(σσ^{T})}{(1 + ϵZ_{t}∥X_{t}∥^{2})^{2}})

≤ k (1 + 𝔼 [\frac{Z_{t}∥X_{t}∥^{2}}{1 + ϵZ_{t}∥X_{t}∥^{2}} ]) .

An application of Gronwall’s inequality (see the next section for a proof of this useful result) establishes,

𝔼 (\frac{Z_{t}∥X_{t}∥^{2}}{1 + ϵZ_{t}∥X_{t}∥^{2}} ) ≤ C(T); ∀t ∈ [0, T].

Applying Fatou’s lemma the desired result is obtained.

Given that Z_{t} has been shown to be a martingale a new probability measure ℙ̃ may be defined by the Radon-Nikodym derivative:

\frac{dℙ̃}{dℙ}|_{F_{T}^{o}} = Z_{T} .

This change of measure allows us to establish one of the most important results in stochastic filtering theory. Proposition 18.3. Under the probability measure ℙ̃, the observations process Y_{t} is a Brownian motion, which is independent of X_{t}. Proof Define a process M_{t} taking values in ℝ by

M_{t} = − ∫_{0}^{t} h^{T} (s, X_{s}) dW_{s},  (11)

is a martingale. Since W_{t} is a ℙ Brownian Motion, which is a martingale under ℙ, it follows that M_{t} is a ℙ local martingale. The previous lemma then establishes that M_{t} is in fact a ℙ martingale. Thus we may apply Girsanov’s theorem, to construct a new measure where

Y_{t} = \frac{dℙ̃}{dℙ}|_{t} = E(M_{t}) = exp (− ∫_{0}^{t} h^{T} (s, X_{s}) dW_{s} − \frac{1}{2} ∫_{0}^{t} ∥h(s, X_{s})∥^{2} ds) = Z_{t}.

As a corollary of Girsanov’s theorem W_{t} − ⟨W, M⟩_{t} is a continuous ℙ̃ local martingale. Lévy’s characterisation theorem then implies that under ℙ̃ the process

W_{t} − ⟨W, M⟩_{t} = W_{t} + ∫_{0}^{t} h(s, X_{s}) ds,

is an m dimensional Brownian Motion. But this process is just the observation process Y_{t} by definition. Thus we have proven that under ℙ̃, the observation process Y_{t} is a Brownian Motion. To establish the second part of the proposition we must prove the independence of Y and X. The law of the pair (X, Y) on [0, T] is absolutely continuous with respect to that of the pair (X, V) on [0, T], since the latter is equal to the former plus a drift term.

Their Radon-Nikodym derivative (which exists since they are absolutely continuous with respect to each other) is Ψ_{T}, which means that for f any bounded measurable function

𝔼_{ℙ} [f(X_{t}, Y_{t})Ψ_{∞}] = 𝔼_{ℙ} [𝔼_{ℙ} [f(X_{t}, Y_{t})Ψ_{t}|F_{t}]]
= 𝔼_{ℙ} [f(X_{t}, Y_{t})𝔼_{ℙ} [Ψ_{∞}|F_{t}]]
= 𝔼_{ℙ} [f(X_{t}, Y_{t})Ψ_{t}] = 𝔼_{ℙ} [f(X_{t}, V_{t})] .

Hence in terms of the probability measure ℙ̃

𝔼ℙ̃ [f(X_{t}, Y_{t})] = 𝔼_{ℙ} [f(X_{t}, Y_{t})Z_{∞}] = 𝔼_{ℙ} [f(X_{t}, V_{t})] ,

and hence under ℙ̃, the variables X and Y are independent.

Proposition 18.4. Let U be an integrable F_{t} measurable random variable. Then

𝔼ℙ̃ [U|Y_{t}] = 𝔼ℙ̃ [U|Y] .

Proof Define

Y′_{t} := σ (Y_{t+u} − Y_{t} : u ≥ 0) ,

then Y = Y_{t} ∪ Y′_{t}. Under the measure ℙ̃, the σ-algebra Y′_{t} is independent of F_{t}, since Y is an F_{t} adapted Brownian Motion which implies that it satisfies the independent increment property. Hence

𝔼ℙ̃ [U|Y_{t}] = 𝔼ℙ̃ [U|Y_{t} ∪ Y′_{t}] = 𝔼ℙ̃ [U|Y] .

18.5. The Unnormalised Conditional Distribution

To simplify the notation in what follows define Z̃_{t} = 1/Z_{t}. Then under the measure ℙ̃ we have

dZ̃_{t} = Z̃_{t}h^{T} (t, X_{t})dY_{t},  (12)

so Z̃_{t} is a ℙ̃ local martingale. Hence

Z̃_{t} = exp (∫_{0}^{t} h^{T} (s, X_{s})dY_{t} − \frac{1}{2} ∫_{0}^{t} ∥h(s, X_{s})∥^{2}ds) .

Also for any stopping time T, 𝔼ℙ̃ \binom{}{Z̃_{T}} = 𝔼_{ℙ} (Z_{T} Z̃_{T} ) = 1, the constant mean condition

implies that Z̃_{T} is a UI ℙ̃ martingale and thus by the martingale convergence theorem Z_{t} → Z_{∞} in L^{1}. So we may write

\frac{dℙ}{dℙ̃}|_{F_{∞}} = Z̃_{∞},

whence

\frac{dℙ}{dℙ̃}|_{F_{t}} = Z̃_{t}, for all t ≥ 0.

Now define, for every bounded measurable function ϕ, the unnormalised conditional distribution of X via

ρ_{t}(ϕ) :=𝔼ℙ̃ [ϕ(X_{t})Z̃_{t}|Y_{t}]
=𝔼ℙ̃ [ϕ(X_{t})Z̃_{t}|Y] ,  (13)

using the result of proposition 18.4.

The following result within the context of stochastic filtering is sometimes called the Kallianpur-Striebel formula. Proposition 18.5. For every bounded measurable function ϕ, we have

π_{t}(ϕ) := 𝔼_{ℙ} [ϕ(X_{t})|Y_{t}] = \frac{ρ_{t}(ϕ)}{ρ_{t}(1)}, ℙ and ℙ̃ a.s..

Proof Let b be a bounded Y_{t} measurable function. From the definition of π_{t} viz

π_{t}(ϕ) := 𝔼_{ℙ} [ϕ(X_{t})|Y_{t}] ,

we deduce from the definition of conditional expectation that

𝔼_{ℙ} (π_{t}(ϕ)b) = 𝔼_{ℙ} (ϕ(X_{t})b) ,

hence by the tower property of conditional expectation

𝔼ℙ̃ (Z̃_{t}π_{t}(ϕ)b) = 𝔼ℙ̃ (Z̃_{t}ϕ(X_{t})b) .

Whence

𝔼ℙ̃ (π_{t}(ϕ)𝔼ℙ̃ [Z̃_{t}|Y_{t}] b) = 𝔼ℙ̃ (𝔼ℙ̃ [ϕ(X_{t})Z̃_{t}|Y_{t}] b) .

which is equivalent to

𝔼ℙ̃ (π_{t}(ϕ)ρ_{t}(1)b) = 𝔼ℙ̃ (ρ_{t}(ϕ)b) .

This implies that

𝔼ℙ̃ (π_{t}(ϕ)ρ_{t}(1)| Y_{t}) = 𝔼ℙ̃ (ρ_{t}(ϕ)| Y_{t})

As the random variables π_{t}(ϕ), ρ_{t}(ϕ) and ρ_{t}(1) from their definitions are all Y_{t} measurable this implies that

π_{t}(ϕ)ρ_{t}(1) = ρ_{t}(ϕ)

Since Z̃_{t} > 0 a.s. which implies that ρ_{t}(1) > 0 a.s. this is sufficient to prove the required result.

Remark The above proof is simply a specific proof of the abstract form of Bayes’ theorem, the general form of which is a standard result, see e.g. A.0.3 in [Musiela and Rutkowski, 2005].

18.6. The Zakai Equation

We wish to derive a stochastic differential equation which is satisfied by the unnormalised density ρ_{t}. Definition 18.6. A set S is said to be total in L^{1}(Ω, F_{t}, ℙ), if for every a ∈ L^{1}(Ω, F_{t}, ℙ)

𝔼 (aϵ_{t}) = 0 ∀ϵ ∈ S ⇒ a = 0.

Lemma 18.7. Let

S_{t} := \{ϵ_{t} = exp (i ∫_{0}^{t} r_{s}^{T} dY_{s} + \frac{1}{2} ∫_{0}^{t} ∥r_{s}∥^{2}ds) : r_{s} ∈ L^{∞} ([0, t], ℝ^{m})\} ,

then the set S_{t} is a total set in L^{1}(Ω, Y_{t}, ℙ̃). The following proof of this standard result is taken from [Bensoussan, 1982]. Proof It suffices to show that if a ∈ L^{1}(Ω, Y_{t}, ℙ̃) and 𝔼̃(aϵ_{t}) = 0 for all ϵ_{t} ∈ S_{t} that a = 0. Take t_{1}, t_{2}, …, t_{p} ∈ (0, t) such that t_{1} < t_{2} < ⋯ < t_{p}, then given k_{1}, …, k_{p}

∑_{j=1}^{p} k_{j}^{T} Y_{t_{j}} = ∑_{j=1}^{p} μ_{i}^{T} (Y_{t_{j}} − Y_{t_{j−1}}) = ∫_{0}^{t} b_{s}^{T} dY_{s}

where

μ_{p} = k_{p}, μ_{p−1} = k_{p} + k_{p−1}, …, μ_{1} = k_{p} + ⋯ k_{1}.

and

b_{s} = \{ \begin{matrix}μ_{j} & for t ∈ (t_{j−1}, t_{j}) \\ 0 & for t ∈ (t_{p}, T) \end{matrix}

Hence

Ẽ (a exp (i ∫_{0}^{t} b_{s} dY_{s})) = 𝔼̃ (a exp (i ∑_{j=1}^{p} k_{j}^{T} Y_{t_{j}} )) = 0

and the same holds for linear combinations by the linearity of 𝔼̃ viz

𝔼̃ (a ∑_{k=1}^{K} c_{k} exp (i ∑_{j=1}^{p} k_{j,k}^{T}Y_{t_{j}} )) = 0

where c_{1}, …, c_{K} are complex coefficients. If F (x_{1}, …, x_{p}) is a continuous bounded complex valued function defined on (ℝ^{m})^{p} then since the set

{f(x_{1}, …, x_{p}) = ∑_{k=1}^{K} c_{k} exp (i ∑_{j=1}^{p} k_{j,k}^{T}x_{j}) : c_{k} ∈ ℂ ∀k, k_{j,k} ∈ ℂ ∀j, k}

is closed under complex conjugation, and separates points in (ℝ^{m})^{p}, the Stone-Weierstrass approximation theorem for complex valued functions implies that there exists a uniformly bounded sequence of functions of the form

P^{(n)}(x_{1}, …, x_{p}) = ∑_{k=1}^{K^{(n)}} c_{k}^{(n)} exp (i ∑_{j=1}^{p} (k_{j,k}^{(n)})^{T} x_{j})

such that

lim_{n→∞} P^{(n)}(x_{1}, …, x_{p}) = F (x_{1}, …, x_{p})

hence

𝔼̃ (aF (Y_{t_{1}}, …, Y_{t_{p}})) = 0

for every continuous bounded F and by the usual approximation argument this extends to every F bounded Borel measurable with respect to σ(Y_{t_{1}}, …, Y_{t_{p}}). As 0 < t_{1} < … < t_{p} < t were arbitrary it follows that 𝔼̃(af) = 0 for every f bounded Y_{t} measurable; taking f(x) = a(x) ∧ n for arbitrary n then implies 𝔼̃(a^{2} ∧ n) = 0, whence a = 0 ℙ̃ a.s..

Remark As a result if ϕ ∈ L^{1}(Ω, Y_{t}, ℙ̃) and we wish to compute 𝔼 (ϕ|Y), it suffices to compute 𝔼 (ϕϵ_{t}) for all ϵ_{t} ∈ S_{t}. Lemma 18.8. Let {U_{t}}_{t≥0} be a continuous F_{t} measurable process such that

𝔼̃ (∫_{0}^{T} U_{t}^{2} dt) < ∞, ∀T ≥ 0,

then for any t ≥ 0, and for j = 1, …, m the following holds

𝔼̃ (∫_{0}^{t} U_{s} dY_{s}^{j}| Y_{t}) = ∫_{0}^{t} 𝔼̃ (U_{s}|Y_{s}) dY_{s}^{j}.

Proof As S_{t} is a total set in L^{1}(Ω, Y_{t}, ℙ̃), it is sufficient to prove that for any ϵ_{t} ∈ S_{t} that

𝔼̃ (ϵ_{t} ∫_{0}^{t} U_{s}dY_{s}^{j}) = 𝔼̃ (ϵ_{t} ∫_{0}^{t} 𝔼̃ (U_{s}|Y_{s}) dY_{s}^{j}) .

To this end note that for every ϵ_{t} ∈ S_{t} the following holds

ϵ_{t} = 1 + ∫_{0}^{t} iϵ_{s}r_{s}^{T} dY_{s},

and hence

\begin{aligned}𝔼̃ (ϵ_{t} ∫_{0}^{t} U_{s}dY_{s}^{j}) & = 𝔼̃ \{(1 + ∫_{0}^{t} iϵ_{s}r_{s}^{T} dY_{s}) (∫_{0}^{t} U_{s}dY_{s}^{j})\} \\ & = Ẽ (∫_{0}^{t} U_{s}dY_{s}^{j}) + 𝔼̃ (∫_{0}^{t} iϵ_{s}r_{s}^{j}U_{s}ds) . \end{aligned}

Here the last term is computed by recalling the definition of the covariation process, in conjunction with the Kunita Watanabe identity. The first term on the right hand side above is a martingale (from the boundedness condition on U_{s}) and hence its expectation vanishes. Thus using the fact that the lebesgue integral and the expectation 𝔼̃(·|Y) commute

𝔼̃ (ϵ_{t} ∫_{0}^{t} U_{s}dY_{s}^{j}) = 𝔼̃ (∫_{0}^{t} iϵ_{s}r_{s}^{j}U_{s}ds)
= 𝔼̃ (𝔼̃ (∫_{0}^{t} iϵ_{s}r_{s}^{j}U_{s}ds| Y))
= 𝔼̃ (∫_{0}^{t} i𝔼̃ (ϵ_{s}r_{s}^{j}U_{s}| Y) ds)
= 𝔼̃ (∫_{0}^{t} iϵ_{s}r_{s}^{j}𝔼̃ (U_{s}|Y) ds)
= 𝔼̃ (∫_{0}^{t} iϵ_{s}r_{s}^{T} dY_{s} ∫_{0}^{t} 𝔼̃ (U_{s}|Y) dY_{s}^{j})
= 𝔼̃ (ϵ_{t} ∫_{0}^{t} 𝔼̃ (U_{s}|Y) dY_{s}^{j}) .

But from proposition 18.4, it follows that since U_{s} is Y_{s} measurable 𝔼̃(U_{s}|Y) = 𝔼̃(U_{s}|Y_{s}) whence

𝔼̃ (ϵ_{t} ∫_{0}^{t} U_{s}dY_{s}^{j}) = 𝔼̃ (ϵ_{t} ∫_{0}^{t} 𝔼̃ (U_{s}|Y_{s}) dY_{s}^{j}) .

As this holds for all ϵ_{t} in S_{t}, and the latter is a total set, by the earlier remarks, this establishes the result.

Lemma 18.9. Let {U_{t}}_{t≥0} be a real valued F_{t} adapted continuous process such that

𝔼̃ (∫_{0}^{T} U_{s}^{2} ds) < ∞, ∀T ≥ 0,  (14)

and let R_{t} be another continuous F_{t} adapted process such that ⟨R, Y ⟩_{t} = 0 for all t. Then

𝔼̃ [∫_{0}^{t} U_{s}dR_{s}|Y_{t}] = 0.

Proof As before use the fact that S_{t} is a total set, so it suffices to show that for each ϵ_{t} ∈ S_{t}

𝔼̃ (ϵ_{t} ∫_{0}^{t} U_{s}dR_{s}) = 0.

As in the previous proof we note that each ϵ_{t} in S_{t} satisfies

ϵ_{t} = 1 + ∫_{0}^{t} iϵ_{s}r_{s}^{T} dY_{s}

Hence substituting this into the above expression yields

𝔼̃ (ϵ_{t} ∫_{0}^{t} U_{s}dR_{s}) =𝔼̃ (∫_{0}^{t} U_{s}dR_{s}) + 𝔼̃ [(∫_{0}^{t} iϵ_{s}r_{s}^{T} dY_{s}) (∫_{0}^{t} U_{s}dR_{s})]
=𝔼̃ (∫_{0}^{t} U_{s}dR_{s}) + 𝔼̃ (∫_{0}^{t} iϵ_{s}U_{s}r_{s}^{T} d⟨Y, R⟩_{s})
=𝔼̃ (∫_{0}^{t} U_{s}dR_{s})
=0.

The last term vanishes, because by the condition 14, the stochastic integral is a genuine martingale. The other term vanishes because from the hypotheses ⟨Y, R⟩_{t} = 0.

Remark In the context of stochastic filtering a natural application of this lemma will be made by setting R_{t} = V_{t}, the stochastic noise process driving the signal process.

We are now in a position to state and prove the Zakai equation. The Zakai equation is important because it is a parabolic stochastic partial differential equation which is satisfied by ρ_{t}, and indeed it’s solution provides a practical means of computing ρ_{t}(ϕ) = 𝔼̃ (ϕ(X_{t})Z̃_{t}|Y_{t}). Indeed the Zakai equation provides a method by which numerical solution of the non-linear filtering problem may be approached by using recursive algorithms to solve the stochastic differential equation. Theorem 18.10. Let A be the infinitesimal generator of the signal process X_{t}, let the domain of this infinitesimal generator be denoted D(A). Then subject to the usual conditions on f, σ and h, the un-normalised conditional distribution on X satisfies the Zakai equation which is

ρ_{t}(ϕ) = π_{0}(ϕ) + ∫_{0}^{t} ρ_{s}(A_{s}ϕ)ds + ∫_{0}^{t} ρ_{s}(h^{T} ϕ)dY_{s}, ∀t ≥ 0, ∀ϕ ∈ D(A). (15)

Proof We approximate Z̃_{t} by

Z̃_{t}^{ϵ} = \frac{Z̃_{t}}{1 + ϵZ̃_{t}} ,

noting that since Z_{t} ≥ 0, this approximation is bounded by

Z̃_{t}^{ϵ} ≤ \frac{4}{27ϵ^{2}} .

Thus we have approximated the process Z_{t} by a bounded process. Application of Itô’s formula to f(x) = x/(1 + ϵx), yields

dZ̃_{t}^{ϵ} = df(Z̃_{t}) = (1 + ϵZ̃_{t})^{−2}Z̃_{t}h^{T} dY_{t} − ϵ(1 + ϵZ̃_{t})^{−3}Z̃_{t}^{2}∥h(t, X_{t})∥^{2}.

The integration by parts formula yields,

d (Z̃_{t}^{ϵ}ϕ(X_{t})) = ϕ(X_{t})dZ̃_{t}^{ϵ} + Z̃_{t}^{ϵ}d (ϕ(X_{t})) + ⟨Z̃_{t}^{ϵ}, ϕ(X_{t})⟩ .

However from the definition of an infinitesimal generator, we know that

ϕ(X_{t}) − ϕ(X_{0}) = ∫_{0}^{t} A_{s}ϕ(X_{s})ds + M_{t}^{ϕ},

where M_{t}^{ϕ} is a martingale given by

M_{t}^{ϕ} = ∑_{j=1}^{d} ∫_{0}^{t} ∑_{i=1}^{d} \frac{∂ϕ}{∂x^{i}} (X_{t})σ_{ij}(t, X_{t}) dW_{t}^{j}.

Using the expression for dZ_{t}^{ϵ} found earlier we may write

d (Z̃_{t}^{ϵ}ϕ(X_{t})) = [Z̃_{t}^{ϵ}Aϕ(X_{t}) − ϕ(X_{t})ϵ(1 + ϵZ̃_{t})^{−3}Z̃_{t}^{2}∥h(t, X_{t})∥^{2}] dt
+ Z̃_{t}^{ϵ}dM_{t}^{ϕ} + ϕ(X_{t})(1 + ϵZ̃_{t})^{−2}Z̃_{t}h^{T} (t, X_{t})dY_{t}

In terms of the functions already defined we can write the infinitesimal generator for X_{t} as

A_{s}ϕ(x) = ∑_{i=1}^{d} f_{i}(s, x)\frac{∂ϕ}{∂x^{i}} (x) + \frac{1}{2} ∑_{i=1}^{d} ∑_{j=1}^{d} (∑_{k=1}^{d} σ_{ik}(s, x)σ_{kj}(s, x)) \frac{∂^{2}ϕ}{∂x^{i}∂x^{j}} (x),

Now we wish to compute

ρ_{t}(ϕ) = 𝔼̃ (Z̃_{t}^{ϵ}ϕ(X_{t})|Y_{t})

we shall do this using the previous expression which on integration gives us

Z̃_{t}^{ϵ}ϕ(X_{t}) =Z̃_{0}^{ϵ}ϕ(X_{0}) + ∫_{0}^{t} [Z̃_{s}^{ϵ}Aϕ(X_{s}) − ϵϕ(X_{t})(1 + ϵZ̃_{s})^{−3}Z̃_{s}^{2}∥h(s, X_{s})∥^{2}] ds
+ ∫_{0}^{t} Z̃_{s}^{ϵ}dM_{s}^{ϕ} + ∫_{0}^{t} ϕ(X_{s})(1 + ϵZ̃_{s})^{−2}Z̃_{s}h^{T} (s, X_{s})dY_{s}  (16)

Note that

𝔼̃ (Z̃_{0}^{ϵ}ϕ(X_{0})| Y_{t}) = 𝔼̃ ( \frac{1}{1 + ϵ}ϕ(X_{0})| Y_{0}) = π_{0}(ϕ)\frac{1}{1 + ϵ}.

Hence taking conditional expectations of both sides of (16)

𝔼̃ (Z̃_{t}^{ϵ}ϕ(X_{t})| Y_{t}) =\frac{π_{0}(ϕ)}{1 + ϵ} + 𝔼̃ [∫_{0}^{t} ϕ(X_{s})(1 + ϵZ̃_{s})^{−2}Z̃_{s}h^{T} (s, X_{s}) dY_{s}| Y_{t}]

+ 𝔼̃ [∫_{0}^{t} (Z̃_{s}^{ϵ}Aϕ(X_{s}) − ϵϕ(X_{s})(1 + ϵZ̃_{s})^{−3}Z̃_{s}^{2}∥h(s, X_{s})∥^{2}) ds| Y_{t}]

+ 𝔼̃ [∫_{0}^{t} Z̃_{s}^{ϵ} dM_{s}^{ϕ}| Y_{t}]

Applying lemma 18.8 since Z̃_{t}^{ϵ} is bounded allows us to interchange the (stochastic) integrals and the conditional expectations to give

𝔼̃ (Z̃_{t}^{ϵ}ϕ(X_{t})| Y_{t}) =\frac{π_{0}(ϕ)}{1 + ϵ} + ∫_{0}^{t} 𝔼̃ (ϕ(X_{s})(1 + ϵZ̃_{s})^{−2}Z̃_{s}h^{T} (s, X_{s})| Y_{t}) dY_{s}

+ ∫_{0}^{t} 𝔼̃ [Z̃_{s}^{ϵ}Aϕ(X_{s}) − ϵϕ(X_{s})(1 + ϵZ̃_{s})^{−3}Z̃_{s}^{2}∥h(s, X_{s})∥^{2}| Y_{t}] ds

+ ∫_{0}^{t} 𝔼̃ [Z̃_{s}^{ϵ}| Y_{t}] dM_{s}^{ϕ}

Now we take the ϵ ↓ 0 limit. Clearly Z̃_{t}^{ϵ} → Z̃_{t} pointwise, and thus by the monotone convergence theorem since Z_{t}^{ϵ} ↑ Z_{t} as ϵ → 0,

𝔼̃ (Z̃_{t}^{ϵ}ϕ(X_{t})|Y_{t}) → 𝔼̃ (Z̃_{t}ϕ(X_{t})|Y_{t}) = ρ_{t}(ϕ),

and also by the monotone convergence theorem

𝔼̃ (Z̃_{s}^{ϵ}A_{s}ϕ(X_{s})|Y_{s}) → ρ_{s}(A_{s}ϕ).

We can bound this by

∀ϵ > 0, 𝔼̃ (Z_{s}^{ϵ}A_{s}ϕ(X_{s})|Y_{s}) ≤ ∥A_{s}ϕ∥𝔼̃ (Z̃_{s}|Y_{s}) ,

and the right hand side is an L^{1} bounded quantity. Hence by the dominated convergence theorem we have

∫_{0}^{t} 𝔼 (Z̃_{s}^{ϵ}A_{s}ϕ(X_{s})|Y_{s}) ds → ∫_{0}^{t} ρ_{s}(A_{s}ϕ) ds a.s.

Also note that

lim_{ϵ→0} ϵϕ(X_{s}) \binom{}{Z̃_{s}^{ϵ}}^{2} (1 + ϵZ̃_{s})^{−1} ∥h(s, X_{s})∥^{2} = 0.

and since the function may be dominated by

|ϵϕ(X_{s}) \binom{}{Z̃_{s}^{ϵ}}^{2} (1 + ϵZ̃_{s})^{−1} ∥h(s, X_{s})∥^{2}| ≤ ϵ∥ϕ∥_{∞}^{2} \frac{ϵZ̃_{s}}{1 + ϵZ̃_{s}} \frac{Z_{s}}{1 + ϵZ̃_{s}} \frac{1}{1 + ϵZ̃_{s}} ∥h(s, X_{s})∥^{2}
≤ Z̃_{s}∥h(s, X_{s})∥^{2}
≤ Z̃_{s}k(1 + ∥X_{s}∥^{2})

But

𝔼̃(Z̃_{s}k(1 + ∥X_{s}∥^{2})) = k𝔼 (Z_{s}Z̃_{s} (1 + ∥X_{s}∥^{2}))
= k (1 + 𝔼(∥X_{s}∥^{2}))
≤ k (1 + C (1 + 𝔼(∥X_{0}∥^{2})) e^{ct})  (17)

Where the final inequality follows by (6). As a consequence of these two results, as the right hand side of the above inequality is integrable, by the conditional form of the Dominated Convergence theorem it follows that

Ẽ (ϵϕ(X_{s}) \binom{}{Z̃_{s}^{ϵ}}^{2} (1 + ϵZ̃_{s})^{−1} ∥h(s, X_{s})∥^{2}| Y) → 0

as ϵ → 0. By proposition 18.4 the sigma field Y may be replaced by Y_{t} which yields that as ϵ → 0

Ẽ (ϵϕ(X_{s}) \binom{}{Z̃_{s}^{ϵ}}^{2} (1 + ϵZ̃_{s})^{−1} ∥h(s, X_{s})∥^{2}| Y_{s}) → 0.

Now note again that the dominating function (17) is also Lebesgue integrable and bounded thus a second application of the Dominated convergence theorem yields

∫_{0}^{t} ϵ𝔼̃ [ϕ(X_{s}) \binom{}{Z̃_{s}^{ϵ}}^{2} (1 + ϵZ̃_{s})^{−1} ∥h(s, X_{s})∥^{2}| Y_{s}] ds → 0, as ϵ → 0.

It now remains to check that the stochastic integral

∫_{0}^{t} 𝔼̃ [ϕ(X_{s})Z̃_{s}^{ϵ} (1 + ϵZ̃_{s})^{−1} h^{T} (s, X_{s})| Y_{s}] dY_{s} → ∫_{0}^{t} ρ_{s}(h^{T} ϕ(X_{s})) dY_{s}.

To this end define

I_{ϵ}(t) := ∫_{0}^{t} 𝔼̃ [ϕ(X_{s})Z̃_{s}^{ϵ} (1 + ϵZ̃_{s})^{−1} h^{T} (s, X_{s})| Y_{s}] dY_{s},

and the desired limit

I(t) := ∫_{0}^{t} ρ_{s}(h^{T} ϕ(X_{s}))dY_{s} = ∫_{0}^{t} 𝔼̃ (Z̃_{s}h^{T} (s, X_{s})ϕ(X_{s})| Y_{t}) dY_{s}.

Both I(t) and I_{ϵ}(t) are continuous bounded square integrable martingales. Consider the difference

I_{ϵ}(t) − I(t) = ∫_{0}^{t} 𝔼̃ [(\frac{Z̃_{s}}{(1 + ϵZ̃_{s})^{2}} − Z̃_{s}) ϕ(X_{s})h^{T} (s, X_{s})| Y] dY_{s}

= − ∫_{0}^{t} 𝔼̃ [ \frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})| Y] dY_{s},

Consider the L^{2} norm of this difference and use the fact that Y is a ℙ̃ Brownian motion

𝔼̃ (I_{ϵ}(t) − I(t))^{2} = 𝔼̃ (∫_{0}^{t} 𝔼̃ [ \frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})| Y] dY_{s})^{2}

= 𝔼̃ (∫_{0}^{t} ∥𝔼̃ [ \frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})| Y]∥^{2} ds)

By the conditional form of Jensen’s inequality ∥𝔼̃(X|Y)∥^{2} ≤ 𝔼̃ (∥X∥^{2}| Y). Thus

∥𝔼̃ [ \frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})| Y]∥^{2}

≤ 𝔼̃ (∥\frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})∥^{2}| Y)

A dominating function can can readily be found for the quantity inside the conditional expectation viz

∥\frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})∥^{2} ≤ ∥ϕ∥_{∞}^{2} |\frac{ϵZ̃_{s}}{1 + ϵZ̃_{s}} (1 + \frac{1}{1 + ϵZ̃_{s}} ) Z̃_{s}|^{2} ∥h(s, X_{s})∥^{2}

≤ 4∥ϕ∥_{∞}^{2}Z̃_{s}^{2}∥h(s, X_{s})∥^{2}
≤ 4k∥ϕ∥_{∞}^{2}Z̃_{s}^{2} (1 + ∥X_{s}∥^{2})

Since

𝔼̃ (Z̃_{s} (Z̃_{s} + Z̃_{s}∥X_{s}∥^{2})) = 𝔼 \binom{}{Z̃_{s}} + 𝔼 (Z̃_{s}∥X_{s}∥^{2})

Using the dominated convergence theorem as ϵ → 0

∫_{0}^{t} ∥𝔼̃ [ \frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}ϕ(X_{s})h^{T} (s, X_{s})| Y_{t}]∥^{2} ds → 0

Now we use the dominated convergence theorem to show that for a suitable subsequence that since ∥h(s, X_{s})∥^{2} ≤ K(1 + ∥X_{s}∥^{2}) we can write

𝔼̃ [∫_{0}^{t} (𝔼̃ [ϕ(X_{s})\frac{ϵZ̃_{s}^{2} (2 + ϵZ̃_{s})}{(1 + ϵZ̃_{s})^{2}}h^{T} (s, X_{s})|Y_{s}])^{2} ds] → 0. a.s.

Thus we have shown that

𝔼̃ [(I_{ϵ}(t) − I(t))^{2}] → 0.

By a standard theorem of convergence, this means that there is a suitable subsequence ϵ_{n} such that

I_{ϵ_{n}}(t) → I(t) a.s.

which is sufficient to establish the claim.

18.7. Kushner-Stratonowich Equation

The Zakai equation which has been derived in the previous section is an SDE which is satisfied by the unnormalised conditional distribution of X_{t}, i.e. ρ_{t}(ϕ). It seems natural therefore to derive a similar equation satisfied by the normalised conditional distribution, π_{t}(ϕ). Recall that the unnormalised conditional distribution is given by

ρ_{t}(ϕ) = 𝔼̃ [ϕ(X_{t})Z̃_{t}|Y_{t}] ,

and the normalised conditional law is given by

π_{t}(ϕ) := 𝔼 [ϕ(X_{t})|Y_{t}] .

As a consequence of the proposition 17.5

π_{t}(ϕ) = \frac{ρ_{t}(ϕ)}{ρ_{t}(1)},

Now using this result together with the Zakai equation, we can derive the Kushner-Stratonovitch equation. Theorem (Kushner-Stratonovich) 18.11. The normalised conditional law of the process X_{t} subject to the usual conditions on the processes X_{t} and Y_{t} satisfies the Kushner-Stratonovich equation

π_{t}(ϕ) = π_{0}(ϕ) + ∫_{0}^{t} π_{s}(A_{s}ϕ)ds + ∫_{0}^{t} (π_{s}(h^{T} ϕ) − π_{s}(h^{T})π_{s}(ϕ)) (dY_{s} − π_{s}(h)ds) . (18)

The process Y_{t} − ∫_{0}^{t} π_{s}(h(s, X_{s})) ds is called the innovations process.

19. Gronwall’s Inequality

An important and frequently used result in the theory of stochastic differential equations is Gronwall’s lemma. There are various versions of Gronwall’s theorem (see [Ethier and Kurtz, 1986], [Karatzas and Shreve, 1987]) but the following general form and proof follows that given in [Mandelbaum et al., 1998].

Theorem (Gronwall). Let x, y and z be measurable non-negative functions on the real numbers. If y is bounded and z is integrable on [0, T], and for all 0 ≤ t ≤ T then

x(t) ≤ z(t) + ∫_{0}^{t} x(s)y(s)ds,  (19)

then

x(t) ≤ z(t) + ∫_{0}^{t} z(s)y(s) exp (∫_{s}^{t} y(r)dr) ds.

Proof Multiplying both sides of the inequality (19) by y(t) exp (− ∫_{0}^{t} y(s)ds) yields

x(t)y(t) exp (− ∫_{0}^{t} y(s)ds) − (∫_{0}^{t} x(s)y(s)ds) y(t) exp (− ∫_{0}^{t} y(s)ds)

≤ z(t)y(t) exp (− ∫_{0}^{t} y(s)ds) .

The left hand side can be written as the derivative of a product,

\frac{d}{dt} [(∫_{0}^{t} x(s)y(s)ds) exp (− ∫_{0}^{t} y(s)ds)] ≤ z(t)y(t) exp (− ∫_{0}^{t} y(s)ds) .

This can now be integrated to give

(∫_{0}^{t} x(s)y(s)ds) exp (− ∫_{0}^{t} y(s)ds) ≤ ∫_{0}^{t} z(s)y(s) exp (− ∫_{0}^{s} y(r)dr) ds,

or equivalently

∫_{0}^{t} x(s)y(s)ds ≤ ∫_{0}^{t} z(s)y(s) exp (∫_{s}^{t} y(r)dr) ds.

Combining this with the original equation (19) gives the desired result.

.

Corollary 19.1. If x, y, and z satisfy the conditions for Gronwall’s theorem then

sup_{0≤t≤T} x(t) ≤ ( sup_{0≤t≤T} z(t)) exp (∫_{0}^{T} y(s)ds) .

Proof From the conclusion of Gronwall’s theorem we see that for all 0 ≤ t ≤ T

x(t) ≤ sup_{0≤t≤T} z(t) + ∫_{0}^{T} z(s)y(s) exp (∫_{s}^{T} y(r)dr) ds,

which yields

sup_{0≤t≤T} x(t) ≤ sup_{0≤t≤T} z(t) + ∫_{0}^{T} z(s)y(s) exp (∫_{s}^{T} y(r)dr) ds,

≤ sup_{0≤t≤T} z(t) + ( sup_{0≤t≤T} z(t)) ∫_{0}^{T} y(s) exp (∫_{s}^{T} y(r)dr) ds

≤ sup_{0≤t≤T} z(t) + ( sup_{0≤t≤T} z(t)) (exp (∫_{0}^{T} y(s)ds) − 1)

≤ ( sup_{0≤t≤T} z(t)) exp (∫_{0}^{T} y(s)ds) .

20. Kalman Filter

It is clear that in the general case the SDEs given by the Kushner-Stratonivich equation are infinite dimensional. A special case in which they are of finite dimension is the most famous example of stochastic filtering is undoubtedly the Kalman-Bucy filter. The relevant equations can be derived without using the general machinery of stochastic filtering theory. The first practical use of the Kalman filter was in the flight guidance computer for the Apollo missions, long before the general non-linear stochastic filtering theory had been developed. However by examining this special linear case, it is hoped that the operation of the Kushner-Stratonowich equation may be made clearer.

We consider the system (X_{t}, Y_{t}) ∈ ℝ^{N} × ℝ^{M} where X_{t} is the signal process and Y_{t} is the observation process, described by the following SDEs,

dX_{t} = (B_{t}X_{t} + b_{t}) dt + F_{t}dV_{t}
dY_{t} = H_{t}X_{t}dt + dW_{t}

where B_{t} and F_{t} are a N × N matrix valued processes, V_{t} is a N dimensional Brownian motion, H_{t} is an M × N matrix valued process and W_{t} is an M dimensional Brownian Motion, which is independent of V_{t}.

Thus in terms of the notation used earlier

\begin{aligned}f(t, X_{t}) & = b_{t} + B_{t}X_{t} \\ σ(t, X_{t}) & = F_{t} \\ h(t, X_{t}) & = H_{t}X_{t} \end{aligned}

The infinitesimal generator of the signal process X_{t} is given by

A_{t}u = ∑_{i=1}^{N} b_{t}^{(i)}(X_{t})\frac{∂u}{∂x_{i}} + \frac{1}{2} ∑_{i=1}^{N} ∑_{j=1}^{N}(F F^{T})_{ij} \frac{∂^{2}u}{∂x_{i}∂x_{j}}

which may readily be verified by applying Itô’s formula to f(X_{t}).

Since the system of equations is linear it is simple to show that the conditional distribution of X_{t} conditional upon Y_{t} is a multivariate normal distribution. Thus it may be characterized by its first and second moments alone. It is therefore sufficient to consider ϕ(X) = x_{i} and ϕ(X) = x_{i}x_{j} as test functions in the Kushner-Stratonovich equation (18). The notation in what follows will become somewhat messy, we write X_{t}^{(i)} for the ith component of the signal process.

Since we shall only be concerned with these two moments, it will simplify the notation to define the conditional mean process

X̂_{t}^{(i)} := π_{t}(x_{i}) = 𝔼 (X_{t}^{(i)}| Y_{t})

and the conditional covariance process

R_{t}^{ij} = 𝔼 ((X_{t}^{(i)} − X̂_{t}^{(i)}) (X_{t}^{(j)} − X̂_{t}^{(j)})) = π_{t}(x_{i}x_{j}) − π_{t}(x_{i})π_{t}(x_{j})

Thus at time t our best estimate of the signal will be a normal distribution N(X̂_{t}, R_{t}).

20.1. Conditional Mean

To derive an equation for the evolution of the conditional mean process, we apply the Kushner-Stratonowich equation to ϕ(X) = X^{(i)}, from theorem 18.11 this satisfies

X̂_{t}^{(i)} = X̂_{0}^{(i)} + ∫_{0}^{t} 𝔼 ((B_{s}X_{s})^{(i)} + b_{t}^{(i)}| Y_{t}) dt
+ ∫_{0}^{t} (𝔼 (X_{s}^{(i)}(H_{s}X_{s})^{T} |Y_{s}) − X̂_{s}^{(i)}(H_{s}X̂_{s})^{T} ) dN_{s}

where N_{t}, the innovations process, is

N_{t} = Y_{t} − ∫_{0}^{t} H_{s}X̂_{s} ds

Whence writing the equation for the evolution of the conditional mean in vector form, we obtain (since R^{T} = R)

dX̂_{t} = (B_{t}X̂_{t} + b_{s}) dt + R_{t}H_{t}^{T} (dY_{t} − H_{t}X̂_{t}dt)  (20)

20.2. Conditional Covariance

We now derive an equation for the evolution of the covariance matrix R, but first we shall need a result about the third moment of a multivariate normal distribution.

If X ∼ N(X̂, R) then

\begin{aligned}𝔼 (X_{t}^{(i)}X_{t}^{(j)}X_{t}^{(k)}) & = 𝔼 ((X̂_{t}^{(i)} + Z^{(i)})(X̂_{t}^{(j)} + Z^{(j)})(X̂_{t}^{(k)} + Z_{t}^{(k)})) \\ & = X̂_{t}^{(i)}X̂_{t}^{(j)}X̂_{t}^{(k)} + X̂_{t}^{(i)}R_{t}^{jk} + X̂_{t}^{j}R_{t}^{ik} + X̂_{t}^{(k)}R_{t}^{ij} \end{aligned}

Applying the Kushner-Stratonovich equation to ϕ(X) = x_{i}x_{j} yields

𝔼 (X_{t}^{(i)}X_{t}^{(j)}| Y_{t}) = 𝔼 (X_{0}^{(i)}X_{0}^{(j)}| Y_{0})
+ ∫_{0}^{t} 𝔼 ((F_{t}F_{t}^{T})^{ij} + (B_{t}X_{t} + b)^{(i)}X_{t}^{(j)} + (B_{t}X_{t} + b)^{(j)}X_{t}^{(i)}| Y_{t}) dt
+ ∫_{0}^{t} {𝔼(X_{t}^{(i)}X_{t}^{(j)}H_{t}X_{t}|Y_{t}) − 𝔼(X_{t}^{(i)}X_{t}^{(j)}|Y_{t})𝔼(H_{t}X_{t}|Y_{t})} · dN_{s} (21)

It is clear that

dR^{ij} = d {𝔼(X_{t}^{(i)}X_{t}^{(j)}|Y_{t})} − d (X̂_{t}^{(i)}X̂_{t}^{(j)})  (22)

the first term is given by (21), and using Itô’s form of the product rule to expand out the second term

d (X̂_{t}^{(i)}X̂_{t}^{(j)}) = X̂^{(i)}dX̂^{(j)} + X̂^{(j)}dX̂^{(i)} + d[X̂^{(i)}, X̂^{(j)}]_{t}

Therefore using (20) we can evaluate this as

d (X̂_{t}^{(i)}X̂_{t}^{(j)}) = X̂_{t}^{(i)}(B_{t}X_{t} + b_{t})^{(j)}dt + X̂_{t}^{(i)}(R_{t}H_{t}^{T} · dN_{t})^{(j)} + X̂_{t}^{(j)}(B_{t}X_{t} + b_{t})^{(i)}dt
+ X̂_{t}^{(j)}(R_{t}H_{t}^{T} · dN_{t})^{(i)} + [(R_{t}H_{t}^{T} · dN_{t})^{(i)}, (R_{t}H_{t}^{T} · dN_{t})^{(j)}]  (23)

For evaluating the quadratic covariation term in this expression it is simplest to work componentwise using the Einstein summation convention and the fact that the innovation process N_{t} is readily shown by Lévy’s characterization to be a ℙ Brownian motion

[(R_{t}H_{t}^{T} · dN_{t})^{(i)}, (R_{t}H_{t}^{T} · dN_{t})^{(j)}] = [R_{t}^{il}H^{kl}dN^{k}, R^{jm}H_{nm}dN^{n}]
= R^{il}H^{kl}R^{jm}H_{nm}δ_{kn}dt
= R^{il}H^{kl}H^{km}R^{jm}dt
= (RH^{T} HR^{T})^{ij}dt
= (RH^{T} HR)^{ij}dt  (24)

where the last equality follows since R^{T} = R. Substituting (24) into (23) and then using this and (21) in (22) yields the following equation for the evolution of the ijth element of the covariance matrix

\begin{aligned}dR_{t}^{ij} & = dt (\frac{1}{2}(F_{t}F_{t}^{T})^{ij} + (B_{t}R_{t})^{ij} + (R_{t}^{T} B_{t}^{T})^{ij} − (RH^{T} HR)^{ij}) \\ & + (X̂_{t}^{(i)}R_{t}^{jm} + X̂_{t}^{(j)}R_{t}^{jm})H_{lm}dN^{l} − (X̂_{t}^{i}R_{t}^{jm} + X̂_{t}^{j}R_{t}^{jm})H^{lm}dN^{l} \end{aligned}

Thus we obtain the final differential equation for the evolution of the conditional covariance matrix; notice that all of the stochastic terms have cancelled

\frac{dR_{t}}{dt} = F_{t}F_{t}^{T} + B_{t}R_{t} + R_{t}B_{t}^{T} − R_{t}H_{t}^{T} H_{t}R_{t}  (25)

The two equations (20) and (25) therefore provide a complete description of the best prediction of the system state X_{t}, conditional on the observations up to time t. This is the Kalman-Bucy filter.

It is most frequently encountered in a discrete time version, since in this form it is practical for use in some of the problems described in the introduction to the stochastic filtering section of these notes.

21. Discontinuous Stochastic Calculus

So far we have discussed integrals with repect to continuous semimartingales. It might seem that this includes a great deal of the cases which are of real interest, but a very important class of processes have been neglected.

The standard Poisson process on the line should be familiar. This has trajectories p(t) which are F_{t}-adapted and piecewise continuous. These have the properties that for any t ≥ 0, p(t) is a.s. finite and for any t, s ≥ 0, p(t + s) − p(t) is independent of F_{t}. The process is also stationary – the distribution of p(t+s)−p(t) does not depend upon t. If the probability of a jump in an infinitessimal time interval is proportional to the length of the time interval, the constant of proportionality is called the intensity λ and it can be shown that the inter jump times have exponential distribution with parameter λ (i.e. mean 1/λ).

Such processes can also be described by their jump measures. The jump measure N(σ, B) is the number of jumps within the time interval σ in the (Borel) set B. From the above definition for any B ∈ B,

𝔼 (N([0, t + s], B) − N([0, t], B) |F_{t}) = λΠ(B),

where Π is a probability measure. More formally, we make the following definition

Definition 1. Poisson Process A Poisson process with intensity measure h(dt, dy) = λ dt×dy as a measurable map N from (Ω, F, ℙ) into the space of positive integer valued counting measures on (ℝ^{+}×Γ, B(ℝ^{+}×Γ)) with the properties

(i) For every t ≥ 0 and every Borel subset A of [0, t] × Γ, N(A) is F_{t−}-measurable. (ii) For every t ≥ 0 and every collection of Borel subsets A_{i} of [t, ∞) × Γ, {N(A_{i})} is

independent of F_{t−}. (iii) E(N(A)) = h(A) for every Borel subset A of ℝ^{+} × Γ. (iv) For each s > 0 and Borel B ⊂ Γ the distribution of N([t, t + s], B) does not depend

upon t.

This measure can be related back to the path p(·) by the following integral for any t ≥ 0,

p(t) = ∫_{0}^{t} ∫_{Γ} γ N(ds, dγ).

The above integral can be defined without difficulty!

21.1. Compensators

Let p(t) be an F_{t}-adapted jump process with bounded jumps, such that p(0) = 0 and 𝔼(p(t)) < ∞ for all t ≥ 0. Let λ(t) be a bounded non-negative F_{t}-adapted process with RCLL sample paths. If

p(t) − ∫_{0}^{t} λ(s) ds

is an F_{t}-martingale, then λ(·) is the intensity of the jump process p(·) and the process ∫_{0}^{t} λ(s) ds is the compensator of the process p(·).

Why are compensators interesting? Recall from the theory of stochastic integration with respect to continuous semimartingales that one of the most useful properties was that a stochastic integral with respect to a continuous local is a local martingale. We would like the same sort of property to hold when we integrate with respect to a suitable class of discontinuous process.

Definition 21.2. For A a non-decreasing locally integrable process, there is a non-decreasing previsible process, called the compensator of A, denoted A^{p} which is a.s. unique characterised by one of the following

(i) A − A^{p} is a local martinagle. (ii) 𝔼(A_{T}^{p}) = 𝔼(A_{T}) for all stopping times T. (iii) 𝔼 [(H · A^{p})_{∞}] = 𝔼 [(H · A)_{∞}] for all non-negative previsible processes H.

Corollary 21.3. For A a process of locally finite variation there exists a previsible locally finite variation compensator process A^{p} unique a.s. such that A − A^{p} is a local martingale.

Example Consider a Poisson process N_{t} with mean measure m_{t}(·). This process has a compensator of m_{t}. To see this note that 𝔼(N_{t} − N_{s}|F_{s}) = m_{t} − m_{s}, for any s ≤ t. So N_{t} − m_{t} is a martingale, and m_{t} is a previsible (since it is deterministic) process of locally finite variation.

Definition 21.4. Two local martingales N and M are orthogonal if their product NM is a local martingale.

Definition 21.5. A local martingale X is purely discontinuous if X_{0} = 0 and it is orthogonal to all continuous local martingales.

The above definition, although important is somewhat counter intuitive, since for example if we consider a Poisson process N_{t} which consists soley of jumps, then we can show that N_{t}−m_{t} is purely discontinuous despite the fact that this process does not consist solely of jumps!

Theorem 21.6. Any local martingale M has a unique (up to indistinguishability) decomposition

M = M_{0} + M^{c} + M^{d},

where M_{0}^{d} = M_{0}^{c} = 0, M^{c} is a continuous local martingale and M^{d} is a purely discontinuous local martingale.

Theorem 21.7. If f is a real valued C^{2} function with domain ℝ^{n} and X is an n-dimensional semi-martingale with decomposition X = M^{c} + M^{d} + Y where M^{c} + M^{d} is a local martingale and M^{c} is a continuous local martingale then the following generalization of Itô’s formula holds

f(X_{t}) =f(X_{0}) + ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} (X_{s−}) dX_{s}

+ \frac{1}{2} ∑_{i,j=1}^{n} ∫_{0}^{t} \frac{∂^{2}f}{∂x_{i}∂x_{j}} (X_{s−}) d[(M^{c})^{i}, (M^{c})^{j}]_{s}

+ ∑_{0<s≤t} (f(X_{s}) − f(X_{s−}) − ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} (X_{s−})ΔX_{s}^{i}) .

The rather ugly notation (M^{c})^{i} refers to the ith component of the continuous local martingale M^{c} and ΔX_{s} = X_{s} − X_{s−}.

A rough heuristic explanation for this generalized form of Itô and an aid to memory of the formula is obtained by noting that the first three terms on the right hand side are identical to those in the multidimensional form of Itô for continuous processes.

Also terms 1,2 and 4 are exactly what you would have expected by writing down an analogue of Itô for discontinuous processes of finite variation. In other words if the process X has a jump at time s, the value of f(X) changes by f(X_{s}) − f(X_{s−}). The jump in X_{s} will give rise to a discontinuity in

∑_{i=1}^{n} ∫_{0}^{t} \frac{∂f}{∂x_{i}} (X_{s−}) dX_{s}

with the jump having size

∑_{i=1}^{n} \frac{∂f}{∂x_{i}} (X_{s−})(X_{s} − X_{s−})

To get the correct jump of f(X_{s}) − f(X_{s−}) then the corrective term is

∑_{0<s≤t} (f(X_{s}) − f(X_{s−}) − ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} (X_{s−})ΔX_{s}^{i}) .

21.2. RCLL processes revisited

We shall be working extensively in the space of right continuous left limits processes (CAD-LAG) and we shall need a metric on the function space. First let us consider the space D_{T} of CADLAG functions from [0, T] to ℝ. The usual approach of using the sup norm on [0, T] isn’t going to be very useful. Consider two functions defined on [0, 10] by

f(x) = \{ \begin{matrix}0 & for x < 1, \\ 1 & for x ≥ 1, \end{matrix}

and

g(x) = \{ \begin{matrix}0 & for x < 1 + ϵ, \\ 1 & for x ≥ 1 + ϵ. \end{matrix}

For arbitrary small ϵ these functions are always one unit apart in the sup metric. We attempt to surmount this problem by considering the class Λ_{T} of increasing maps from [0, T] to [0, T]. Now define a metric

d_{T} (f, g) := inf {ϵ : sup_{0≤s≤T} |s − λ(s)| ≤ ϵ, sup_{0≤s≤T} |f(s) − g(λ(s))| ≤ ϵ for some λ ∈ Λ_{T} } .

This metric would appear to nicely have solved the problem described above. Unfortunately it is not complete! This is a major problem because Prohorov’s theorem (among others) requires a complete separable metric space! However the problem can be fixed via a simple trick; for a time transform λ ∈ Λ_{T} define a norm

|λ| := sup_{0≤s≤T} |log \{\frac{λ(t) − λ(s)}{t − s}\}| .

Thus we replace the condition |λ(s) − s| ≤ ϵ by |λ| ≤ ϵ and we obtain a new metric on D_{T} which this time is complete and separable;

d′_{T} (f, g) := inf {ϵ : |λ| ≤ ϵ, sup_{0≤s≤T} |f(s) − g(λ(s))| ≤ ϵ, for some λ ∈ Λ_{T} } .

This is defined on the space of CADLAG functions on the compact time interval [0, T] and can be exteneded via another standard trick to D_{∞},

d′(f, g) := ∫_{0}^{∞} e^{−t} min [1, d′_{t}(f, g)] dt.

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