P R O B A B I L I T Y,    S p r i n g   2 0 2 5

Lectures: Tuesday, Thursday, 2:00-3:15pm, in Warren Weaver Hall 517.

Lecturer: Paul Bourgade, office hours Wednesday 11.00am-12.00, you also can email me (bourgade@cims.nyu.edu) to set up an appointment or just drop by (WWH 629).

Course assistant: Douglas Dow (cdd3103@nyu.edu).

Course description: The course is targeted at Mathematics PhD students. Stochastic processes in continuous time. Brownian motion. Poisson process. Processes with independent increments. Stationary processes. Semi-martingales. Markov processes and the associated semi-groups. Connections with PDEs. Stochastic differential equations. Convergence of processes.

Prerequisites: Probability 1, i.e. the content of the book Probability Theory by S.R.S. Varadhan.

Textbooks: Our reference text will be Stochastic Processes, by S.R.S. Varadhan.

Homework: Every Tuesday for the next Tuesday.

Grading: problem sets (50%) and final (50%).

A tentative schedule for this course is:

• Jan 21. 1.1 Continuous time processes.
• Jan 23. 1.2 Continuous parameter martingales.
• Jan 28. 1.3 Semimartingales. 1.4 Martingales and stochastic integrals.
• Jan 30. 2.1 The basic Poisson process. 2.2 Compound Poisson process.
• Feb 4.
• Feb 6.
• Feb 11.
• Feb 13.
• Feb 20.
• Feb 15.
• Feb 27.
• Mar 4.
• Mar 6.
• Mar 11.
Mar 13.
• Mar 18.
• Mar 20.
• Apr 1.
• Apr 3.
• Apr 8.
• Apr 10.
• Apr 15.
• Apr 17.
• Apr 22.
• Apr 24.
• Apr 29.
• May 1.
• May 6.

• Problem set 1: Exercises 1.1, 1.2, 1.6, 2.2, 2.3.