**
MATH 286: STOCHASTIC DIFFERENTIAL EQUATIONS (FALL 1999)
**

** Professor: **
Professor R. J. Williams, email: williams@math.ucsd.edu.

** Office: ** AP&M 6121. ** Office hours: **
W, F 1-1.50 p.m. and by appointment.

** Lecture time: ** MW, 4.40-6 p.m. ** Place: ** HSS 2305B.

** First class meeting: ** Monday, October 4, 1999.

**
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.
** 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.

** TEXT: **

Oksendal, B., Stochastic Differential Equations,
Springer, 5th edition, 1998.
** RECOMMENDED ADDITIONAL REFERENCES: **

Chung, K. L., and Williams, R. J., Introduction to Stochastic
Integration, Second Edition, Birkhauser, 1990.
Revuz, D., and Yor, M., Continuous Martingales and Brownian Motion,
Springer, Third Edition, 1999.
** OTHER REFERENCES: **

* Theory*

Karatzas, I. and Shreve, S., Brownian motion and stochastic
calculus, 2nd edition, Springer.
Metivier, M., Semimartingales, de Gruyter, Berlin, 1982.
Protter, P., Stochastic Integration and Differential
Equations, Springer.
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.
** SOFTWARE **

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).
** RECOMMENDED PREREQUISITE: **

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