MATH 275D: Winter 2024

Stochastic Calculus

lecturer: Marek Biskup, MS 6180
lecture: MWF 11-noon in MS 7608
discussion: (tentatively) R 11-noon in TBD, aimed at HW problem solving, etc
office hours: (tentatively) MWF 10-11 or by appointment

Take-home Final Exam:  download in PDF

Homework:

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

Lectures:

Here you will find board photos & lecture notes along with a brief description of material covered.

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

• Dynamical Theories of Brownian Motion by E. Nelson
• Brownian Motion by P. Mörters and Y. Peres

General plan: This class is a formal continuation of the Graduate Probability (MATH 275ABC) sequence. Here is an outline of what I plan to cover:

• A quick intro into Brownian motion and continuous-time stochastic processes
• Stochastic (Ito and Stratonovich) integral and Ito calculus
• Additive and multiplicative chaos theory
• Stochastic differential equations, Ito diffusions
• Applications to harmonic analysis, filtering, control theory, economics

Prerequisites:

The course will be based on measure theory and Lebesgue integration so working knowledge of these is required. Similarly, I will also expect proficiency in the formalism and notation of probability and will use without apology the concepts of conditioning and independence (for both random variables and sigma algebras). 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: The first class (Monday, Jan 8) will take place over zoom sent to class roster (use above email to request the link if not enrolled). Physically, we will first meet on Wednesday, Jan 10. The class will be finished by a take-home final exam to be assigned during the last week of instruction.

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 on in class on the due date indicated.