**
MATH 286: STOCHASTIC DIFFERENTIAL EQUATIONS (FALL 2016)
**

** 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.

**
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.
** 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.

** RECOMMENDED ADDITIONAL REFERENCES: **

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.
** OTHER REFERENCES: **

* Theory *

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.
** RECOMMENDED PREREQUISITE: **

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

Last updated April 17, 2016.