This is an introduction to stochastic calculus and stochastic differential equations, emphasizing the connections with partial differential equations and applications.
I plan to follow (roughly)
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:
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:
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.
I plan to record the in-person lectures using Zoom and make them available in Canvas.
I will follow the University’s academic integrity policy. If you have any concerns or questions about this policy or academic integrity in class, please contact me.
Please include Math 4971 in the subject line of any email message that pertains to this course.