Professor: Professor R. J. Williams
Office: AP&M 6121.
Office hours: M 1-1.50 p.m., W 3-3.50 p.m.

Lecture time: MW 4-5.20 p.m. Note that this is changed from the original scheduled time.

Place: AP&M 6218.

DESCRIPTION: Stochastic differential equations arise in modelling a variety of random dynamic phenomena in the physical, biological, engineering and social sciences. Solutions of these equations are often diffusion processes and hence are connected to the subject of partial differential equations. This course will present the basic theory of stochastic differential equations and provide examples of its application.


  • 1. A review of the relevant stochastic process and martingale theory.
  • 2. Stochastic calculus including Ito's formula.
  • 3. Existence and uniqueness for stochastic differential equations, strong Markov property.
  • 4. Applications.

    COMPUTER MODULES: Computer modules for performing symbolic manipulations in stochastic calculus and for numerically approximating the solutions of stochastic differential equations will be made available to complement the theoretical material presented in the course. These modules make use of Mathematica.


  • Chung, K. L., and Williams, R. J., Introduction to Stochastic Integration, Second Edition, Birkhauser, 1990.


  • Karatzas, I. and Shreve, S., Brownian motion and stochastic calculus, 2nd edition, Springer.
  • Oksendal, B., Stochastic Differential Equations, Springer, 5th edition, 1998.


  • Metivier, M., Semimartingales, de Gruyter, Berlin, 1982.
  • Protter, P., Stochastic Integration and Differential Equations, Springer.
  • Revuz, D., and Yor, M., Continuous Martingales and Brownian Motion, Springer, Third Edition, 1999.
  • Rogers, L. C. G., and Williams, D., Diffusions, Markov Processes, and Martingales, Wiley, Volume 1: 1994, Volume 2: 1987.
  • Jacod, J., and Shiryaev, A. N., Limit theorems for stochastic processes, Springer-Verlag, 1987.
    Numerical solution of SDEs
  • Pardoux, E., and Talay, D., Discretization and simulation of stochastic differential equations, Acta Applicandae Mathematicae, 3 (1985), 23--47.
  • Talay, D., and Tubaro, L., Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Analysis and Applications, 8 (1990), 94-120.
  • Kloeden, P. E., and Platen, E., Numerical solution of stochastic differential equations, Springer, 1992.
  • Kloeden, P. E., Platen, E., and Schurz, H., Numerical solution of SDEs through computer experiments, Springer, 1994.
  • Bouleau, N., and Lepingle, D., Numerical Methods for Stochastic Processes, Wiley, 1994.
  • Gaines, J. G., and Lyons, T. J., Variable step size control in the numerical solution of stochastic differential equations, SIAM J. Applied Math., to appear.


  • Symbolic Stochastic Calculus Software developed by Wilfrid Kendall, University of Warwick. This runs under Mathematica for example.
  • C programs for the numerical solution of stochastic differential equations, provided by Jessica Gaines, Edinburgh University, U.K.
  • Software for numerical study of the stochastic Brusselator (provided by Gabriele Bleckert and Klaus Reiner Schenk-Hoppe).

    Math 280AB (Probability) or equivalent, or consent of instructor. Please direct any questions to Professor Ruth J. Williams, email:

    Last updated September 24, 2003.