MATH 275D: Fall 2025

MATH 275D: Fall 2025

Stochastic Calculus

lecturer: Marek Biskup, MS 6180
lecture: MWF 10-11 in MS 7608
discussion: R 10-11 in MS 6201
office hours: (tentatively) MWF 11-noon or by appointment

Take-home Final Exam: download in PDF

(due Saturday Dec 20, 11:59PM Pacific Time)

Homework:

HW#1 HW#2 HW#3 HW#4 HW#5 HW#6 HW#7

Lecture summary, board photos & notes: see here

Resources: The principal textbooks for this class are

Brownian Motion and Stochastic Calculus by Y. Karatzas and S.E. Shreve
Stochastic Differential Equations by B. Øksendal

Further resources and supplementary materials:

Dynamical Theories of Brownian Motion by E. Nelson
Brownian Motion by P. Mörters and Y. Peres
Continuous time Markov processes by T.M. Liggett

General plan: This class is a vague continuation of the Graduate Probability (MATH 275ABC) sequence. Here is an outline of what will be covered:

Introduction to stochastic processes focussed primarily on Brownian motion
Stochastic (Ito and Stratonovich) integral and Ito calculus
Stochastic differential equations, Ito diffusions
Solving PDEs using stochastic processes
Additive and multiplicative chaos theory and applications to SDEs

The remaining topics are optional and dependent on time and/or interests:

Rough paths and associated integration theory
Applications to harmonic analysis, filtering, control theory, economics

Note: The class will be focused on mathematics so no prior understanding of the applications will be needed.

Prerequisites:

The course will be based on measure theory and Lebesgue integration so working knowledge of these is required. Similarly, proficiency in the formalism and notation of probability will be expected; the concepts of conditioning and independence (for both random variables and sigma algebras) will be used without apology. We will also sometimes invoke facts about asymptotic laws for sums of independent random variables (the SLLN, CLT, etc) as well as various slick convergence theorems from the theory of discrete time Markov chains and martingales. All the needed prereqs will still be reviewed but the pace may be fast.

Scheduling: Due to conference travel of the lecturer, the classes on Oct 10, 13, 15 and 17 will be canceled and rescheduled to a later date. One way is to create an extra hour during the week when we can meet for informal discussion. Details will be discussed during the first week of classes.

Formal matters: There will be regular homework assignments and a take-home final exam with the two categories contributing equally to the final grade. Letter grading (i.e., A, B+, C-, etc.) will be used based with equal proportion on homework and the final. The homework assignments will be distributed though this website and are to be turned in by email in PDF format on the stated due date.