Professor: Professor R. J. Williams
Email: williams "at" math "dot" ucsd "dot" edu.
Office: AP&M 7161.
Office hours: Tu, noon-12.50pm., Th 3-3.50 p.m.

Lecture time: TuTh 5-6.30 p.m. (There will be no class on Tuesday, November 1, 2016. This class will be made up with extra time for the other class meetings.)

Place: AP&M 5402.

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.

    SOME NOTES: You will need the information given in class to access this section.

    HOMEWORK: Click here

    TEXT: Introduction to Stochastic Integration, K. L. Chung and R. J. Williams, Birkhauser, Boston, second edition, 1990.
    For a list of errata, click here.


  • Oksendal, B., Stochastic Differential Equations, Springer.
  • Karatzas, I. and Shreve, S., Brownian motion and stochastic calculus, 2nd edition, Springer.
  • Kloeden, P. E., Platen, E., and Schurz, H., Numerical solution of SDEs through computer experiments, Springer, Second edition, 1997.


  • Baudoin, Fabrice, Diffusion Processes and Stochastic Calculus, European Math. Society (available from the American Math Society) 2014.
  • 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, Second edition, 2003.
    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.
  • 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., 57 (1997), no. 5, 1455-1484.

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

    Last updated April 17, 2016.