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. 2.3 Infinite number of small jumps.
• Feb 6. 2.3 Infinite number of small jumps.
• Feb 11. 2.4 Infinitesimal generators. 2.5 Some associated martingales.
• Feb 13. 3.1 Point processes. 3.2 Poisson point processes.
• Feb 20. 4.1 Simple examples
• Feb 25. 4.2 Semigroups of operators.
• Feb 27. 4.3 Example: Birth and Death process. 4.4 Markov processes and martingales
• Mar 4. 4.5 Explosion.
• Mar 6. 4.6 Recurrence, transcience. 4.7 Invariant distributions.
• Mar 11. 5.1 Definition of Brownian motion, construction by Levy.
• Mar 13. 5.1 Construction based on Ito-Nisio, Holder regularity.
• Mar 18. Invariance properties.
• Mar 20. Strong Markov property, reflection principle.
• Apr 1. Skorokhod embedding,
• Apr 3.
• Apr 8.
• Apr 10.
• Apr 15.
• Apr 17.
• Apr 22.
• Apr 24.
• Apr 29.
• May 1.
• May 6.

• Problem set 1, due Feb 4: Exercises 1.1, 1.2, 1.6, 2.2, 2.3.
• Problem set 2, due Feb 11: Exercises 2.5, 2.6, 2.7.
• Problem set 3, due Feb 25: Exercises 3.1, 3.2, 3.3, 3.4.
• Problem set 4, due Mar 4: Exercises 4.2, 4.3, 4.4, 4.5.
• Problem set 5, due Mar 11: Exercises 4.6, 4.7, 4.8, 4.10, 4.12, 4.13.
• Problem set 6, click here, tex file due Apr 1.