Alan Bain
I can be contacted as alanb<at>chiark.greenend.org.uk.
Stochastic Calculus and Stochastic Filtering
This is the new home for a set of stochastic calculus notes
which I wrote which seemed to be fairly heavily used. They used to
be based on a University of Cambridge server.
Stochastic Calculus Notes
These notes provide a fairly complete elementary introduction to the basics of
stochastic integration with respect to continuous semimartingales (not just
with respect to a Brownian Motion). They contain all the theory
usually needed for basic mathematical finance (Girsanov's theorem, Brownian
Martingale Representation Theorem).
The Stochastic filtering section provides an elementary introduction
to this subject beginning from the viewpoint of non-linear
filtering extending as far as the Zakai equation and the
Kushner-Stratonowich equation.
The more usual starting point of the
Kalman-Bucy linear stochastic filter is derived as an example
of the general non-linear theory.
The notes may be downloaded
and print out at about one hundred pages. If you find any errors, or feel that
there are serious omissions, or even just have some suggestions for
improvements, please contact me by email and I shall
endeavour to improve them!
Last Updated: 18th June 2007
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 in L and M in 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. Ito's Formula 35
11.1. Applications of Ito's Formula 40
11.2. Exponential Martingales 41
12. Levy 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-Kac 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 93
22. References 95
The notes are available in various forms, but I have had reports of people
experiencing trouble with the postscript versions. While a PDF version is
now to be expected, the original idea for a PDF version of these notes
was suggested to me by Noel Vaillant at a time when PDF usage was much less
common.
Download (Version of 25th May 2009)
Recent additions include: corrections to the path regularization
theorem, an example of a local martingale which is not
a martingale, existence and uniqueness of strong solutions of
SDEs with lipshitz coefficient, an expanded section on exponential
martingales, compensators of discontinuous processes.
The Percolation Phase Transition (1999)
In percolation models lots of interesting behaviour takes place in the
vicinity of the critical point, about which very little is known in
some quite simple models. One of the pieces of definate information
is provided by the work of Hara and Slade on the Lace Expansion.
My part III essay provides what aims to be a simple overview of the lace
expansion and what it achieves. It may at the moment only be downloaded
as postscript because it uses some `home-made' metafonts, and I am unsure
of the best way to distribute these for viewing by other people.
Also of interest may be the appendix which contains some Monte Carlo
simulations to demonstrate conformal invariance and Cardy's formula
for site percolation on a square and a triangular lattice.
Numerical C++
The following is a list of C++ numerical code which I have
written which may potentially be of use to other people. It's aim
has been to explore simple and efficient ways to make use of C++
in numerical code. The routines may be used in other programs
provided the copyright notice is retained intact.
- Non-linear root finder (algorithm
J.C.P. Bus & T.J. Dekker "Two Efficient Algorithms with Guaranteed Convergenece
for finding a Zero of a Fuction" in ACM Trans. on Math. Software (TOMS)
Vol 1., No. 4 Dec 1975 pages 330-345.
- bd.h main routine (templated)
- bdmain.cpp example driver (solves quadratic)
Example Tidal Height Calculation
As a demonstration of how tidal heights may be computed from their
harmonic components I wrote a simple FORTRAN program
tide.f with some example
components for the port of Liverpoot port.data.
Lathe Work
- Postscript/PDF to GCODE Translator Convert
your postscript files to GCODE suitable for controlling a CNC
milling machine
- Notes on my lathes, which include a Hardinge HLV-H toolroom lathe,
two Lorch Schmidt 4in plain lathes and two watchmaker's lathes (a WW pattern 8mm lathe and a D-bed type 6mm which is mainly
used for between centres work).
- Martin Screw Threads
Radio
The following are links to information about old radio communications
equipment which I have worked on in the past.
Old Computer Information
HP 34401A DVM
One of the best bench DMMs (in my personal opinion) is the HP34401A. T
here's not much information around on how it works and it has a rather
nice diagnostic interface over HPIB, so I've been gradually writing up
some of my notes on HP34401A operation.