## Math 4971 – Stochastic Differential Equations (Fall 2021)

### Section information

• Class time and location: M-W-F from 1:00PM to 1:50PM in Ridgley 107
• Tentative office hours: Monday-Wednesday-Thursday 2:00PM to 4:00PM (shared)

### Subject

This is an introduction to stochastic calculus and stochastic differential equations, emphasizing the connections with partial differential equations and applications.

### Text

• An Introduction to Stochastic Differential Equations by Lawrence C. Evans, American Mathematical Society, 2013.

This text presupposes knowledge of measure theory. My plan is to spend the early part of the course introducing (or reviewing, for those already familiar with) the essentials of probability theory based on a measure theoretic/Lebesgue integral foundation. There are many stochastic processes books that cover this material. For example, the first two chapters of the following text provides a helpful outline very much like what I intend to cover:

• Stochastic Modeling by Nicolas Lanchier. Universitext, Springer, 2017. Library link here.

One problem with Evans text is the dearth of exercises. For homework assignments, I may take problems from a variety of texts. The following may be a useful source of interesting problems:

• Stochastic Differential Equations: An Introduction with Applications by Bernt Øksendal. Universitext, Springer 2014.

### Topics we hope to cover.

• Measure and probability: Lebesgue integral for general measure spaces, probability spaces, conditional expectation

• The main limit theorems in probability, martingales

• Brownian motion: motivation, construction, the Markov property

• Stochastic integrals and Itô calculus

• Stochastic differential equations

• Applications to PDEs, diffusions, the Feynman-Kac formula, stochastic Petri nets

Coursework will be limited to homework assignments. I expect to give about 5 or 6 in number. Final grades will be based entirely on homework grades and class attendance. Further details will be given in class. Assignments will be collected through Crowdmark.

### Recording

I plan to record the in-person lectures using Zoom and make them available in Canvas.