Last Updated: 18th June 2007

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.

- Level 2 Postscript
- Gzipped Postscript
- DVI
- Short Version PDF (without Stochastic Filtering/Discontinuous Processes

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.

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.

- 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)

**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**

- Wireless (MAS90)
- Eddystone EC958
- Old Tektronix Scopes
- Alan Clayton's (G7HZZ) Eddystone Site
- Creed 7 Teleprinter
- Test Equipment

- Sun Hardware FAQ (1995)
- Sun Hardware FAQ (1997)
- Copy of the Sun Hardware FAQ (being updated, principally to include more complete information on the older hardware, rather than to include information on the newer hardware)
- Sony Monitor Adjustment Information
- SunHELP a useful collection of Sun resources
- RS 232 Information
- Laserwriter IINT set up information
**Linux Emerald QIC Driver**I wrote some years ago an Emerald Interface Support into the Linux QIC-02 driver. This is available from chiark's ftp site. This was developed on a 1.3.49 kernel, which is fairly aged by today's standards. I wouldn't be surprised if it didn't work at all any longer. QIC-02 driver source

- The R Language for statistical data analysis Documentation
- The PARI computer algebra system Bordeaux FTP site.