Math 286 Fall 2022 |
Stochastic Calculus and Stochastic Differential Equations
This class is an introduction to stochastic integration and stochastic differential equations for continuous semimartingales. Expected background includes the contents of the Math 280 full course in graduate probability. In particular, students should already be familiar with the basics of stochastic processes (filtrations, stopping times, quadratic variation, Brownian motion and other continuous Markov processes) and martingales (optional stopping, local martingales, Doob-Meyer decomposition). After briefly reviewing these topics, we will develop the stochastic integral with respect to Brownian motion, and then generalize this to predictable processes. We then proceed to Ito's formula, and the standard theory of existence and uniqueness for stochastic differential equations (SDEs). Time permitting, we will conclude with a brief discussion of white noise and stochastic partial differential equations (SPDEs). Text Resources:
The Lectures will typically be given on an iPad. The before and after Lectures Slides will be posted on Canvas. There, you will also find the video podcast/screencast recordings of the lectures; those can also be found at podcast.ucsd.edu. Grades: There will be 3 homework sets. They will be posted below and on Canvas, with due dates, when they are available. You should turn in the homework through Gradescope. Homework 1: due on Sunday, October 23, by 11:59pm. Homework 1. Homework 2: due on Sunday, November 13, by 11:59pm. Homework 2. Homework 3: due on Friday, December 9, by 11:59pm. Homework 3. |