MATH 286: STOCHASTIC DIFFERENTIAL EQUATIONS (FALL 2005)

Professor: Professor R. J. Williams
Email: williams "at" math "dot" ucsd "dot" edu.
Office: AP&M 6121.
Office hours: M, W 5.30-6.20 p.m.

Lecture time: MW 4-5.20 p.m. The first class meeting will be on Monday, September 26, 2005. There will be NO class meeting on Monday, October 10, 2005, only. This class will be made up at another time.

Place: AP&M 7421

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.

TOPICS:

  • 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.

    HOMEWORK: Click here

    TEXT:

  • Oksendal, B., Stochastic Differential Equations, Springer, 6th edition.

    RECOMMENDED ADDITIONAL REFERENCES:

  • 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.
  • Kloeden, P. E., Platen, E., and Schurz, H., Numerical solution of SDEs through computer experiments, Springer, Second edition, 1997.

    OTHER REFERENCES:
    Theory

  • 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.
  • 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.

    SOFTWARE

  • Sasha Cyganowski, Peter E. Kloeden and Jerzy Ombach, From Elementary Probability to Stochastic Differential Equations with MAPLE, Springer Verlag (2001) and Peter E. Kloeden, Eckhard Platen and Henri Schurz, Numerical Solution of SDE Through Computer Experiments, Springer Verlag (1994, Second printing 1997).
  • The MAPLE Stochastic Package, by Sasha Cyganowski, maintained by Lars Grune.

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

    Last updated September 25, 2005.