Math 286     Fall 2022

Stochastic Calculus and Stochastic Differential Equations

  name e-mail lecture / office hours office
Instructor: Todd Kemp tkemp@ucsd.edu MWF 10:00-10:50a in AP&M 2402 AP&M 5202
TA: Vaki Nikitopoulos enikitop@ucsd.edu Th 9:30-11:00a in HSS 4072 or HSS 4012 HSS 4072

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 main text resource we will follow for the course are the Lecture Notes produced by Vaki Nikitopoulos from a previous iteration of this course, found here.
  • These notes partially follow the development in the textbook Intorduction to Stochastic Integration (Second Edition) by Chung and Williams; it is available at the UCSD bookstore, and also free to UCSD affiliates as a pdf download from SpringerLink.
  • The recent textbook "An Introduction through Theory and Exercises" by Baldi is a good alternate source for this material. It is available for free to UCSD affiliates as a pdf download from SpringerLink.
  • Seppalainen's very polished notes give a more comprehensive treatment than we need: Basics of Stochastic Analysis.
  • For the necessary background in measure theory, probability theory, and stochastic processes, please see my year-long YouTube course.

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.