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.

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

Radio

The following are links to information about old radio communications equipment which I have worked on in the past.

Old Computer Information

Mathematical Software

These are some useful readily available programs available for download
The R Language for statistical data analysis Documentation
The PARI computer algebra system Bordeaux FTP site.