Math 735 Stochastic Differential Equations

Instructor:	Tom Kurtz
Prerequisite:	Math 431 AND consent of instructor. Consent is automatic for anyone who has had Math 629, 721, or 831 or Stat 709.
Textbook:	Notes will be provided. The old version is available at: http://www.math.wisc.edu/~kurtz/m735.htm

The course will introduce stochastic integrals with respect to general semimartingales, stochastic differential equations based on these integrals, integration with respect to Poisson random measures, stochastic differential equations for general Markov processes, change of measure, and applications to finance, filtering and control.

The standard, but challenging, reference is Stochastic Integration and Differential Equations. A New Approach by Philip Protter. Copies will be available in the Bookstore.

Course Outline

I. A brief review of probability

1. Basic concepts

2. Random variables and expectations

3. Representation of information obtained by observations

4. Conditional expectation and probability

5. Basic concepts of stochastic processes

6. The Poisson process and Brownian motion

II. Stochastic integration

1. Stieltjes integral

2. Definition of stochastic integral

3. Existence of stochastic integral for finite variation processes

4. Definition and properties of martingales

5. Existence of stochastic integral for martingales

6. Ito's formula

III. Stochastic differential equations

1. Existence and uniqueness

2. Approximation theorems

IV. Markov processes

1. Ito equations for diffusion processes

2. Poisson random measures

3. Stochastic differential equations for Markov processes with jumps

4. Exit times

5. Reflected diffusion processes

V. Change of measure

1. Radon-Nikodym theorem

2. Bayes formula

3. Martingales under a change of measure

4. Change of measure for Poisson processes and Brownian motion

VI. Examples and applications

1. Filtering

2. Finance

3. Control

4. Backward stochastic differential equations