Math 247A (Driver, Winter 2009) Topics in Real Analysis

Rough Path Analysis

(http://math.ucsd.edu/~driver/247A-Winter2009/index.htm)

Instructor: Bruce Driver (bdriver@math.ucsd.edu), AP&M 7414, 534-2648.

Office Hours: TBA

Meeting times: Lectures are on MWF 11:00a - 11:50a in AP&M B412.

Textbook: There is no official text book for this course. However, there will likely be posted lecture notes on the course web-site.  Other references will be supplied as the course progresses.

Prerequisites:
A standard undergraduate course in real analysis.  For later parts of the course, some knowledge of measure theory and probability theory would be helpful.

Course Description: This course will be concerned with Terry Lyons’ theory which is called Rough Path Analysis.  The theory is devoted to solving ordinary differential equations driven by nowhere differentiable paths.  The typical equation is of the form;

dXt = A(Xt)dBt with X0 = x0 .

In this equation Bt is assumed to be a rough path with infinite variation.  One of the motivations of this theory is to give a deterministic interpretation of stochastic differential equations where the typical choice for Bt is a Brownian motion.  (The notion of Brownian motion is not a prerequisite for this course.)  In the case of Brownian motion, Bt  has infinite variation and hence the classical Stieltjes integration theory does not apply here.  Nevertheless, building on the work of Young, Chen and others, Terry Lyons has developed techniques to handle such rough equations.  In order to make this theory work, one must “augment” the driving noise, Bt  by its “Levy area” process.  The goal of this course will be to describe this theory and give some applications of it to stochastic differential equations.

List of Possible Topics

 A warm up with Young’s integral and ordinary differential equations for not so rough paths. The notions of rough(er) paths and there augmentations. Rough path integration theory. Solving rough path ordinary differential equations by both Picard iterates and Peano's techniques. Brownian motion and other processes as examples of rough paths. Numerical techniques involving rough paths. Malliavin/Hormander density results for stochastic differential equations driven by fractional Brownian motions. Applications to stochastic partial differential equations.

A Few Suggested References

References 1. and 3. may be acquired via MathSciNet through your UCSD account.