**
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;

*dX*_{t} = A(X_{t})dB_{t
}with *X*_{0 }= x_{0 }.

In this equation *B*_{t }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 *B*_{t }is a Brownian motion. (The notion of
Brownian motion is not a prerequisite for this course.) In the case of Brownian
motion, *B*_{t }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, *B*_{t }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**