### Information

Mondays, 1:25-3:15pm, Warren Weaver Hall (Courant Institute) Location: WWH 312

Instructor: Miranda Holmes-Cerfon

Office Hours: Thursdays 3:30-5:30pm, WWH 1107

Prerequisites: basic knowledge (e.g. beginning graduate) of: probability, linear algebra, ODEs, PDEs, analysis, perturbation methods, plus a general graduate level of mathematical sophistication. You should have taken the graduate-level "Basic Probability" course here at Courant or have equivalent knowledge. A full course on measure theory is not required but elements of it will occasionally be referred to.

Homework: Due Mondays in class. Download from NYU Classes website.

Grading will be based on weekly homework assignments, and (probably) a final exam. The exam will given roughly the weight of two homework assignments.

This is a graduate class aimed at beginning PhD students in applied mathematics, that will introduce the major topics in stochastic analysis from an applied mathematics perspective. Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, Fokker-Planck equation, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments.

Information from last year's course is here.

### References

The recommended textbook for the course is-
*Stochastic Processes and Applications*, by G. A. Pavliotis.

For the first few weeks, it will be helpful to look at a reference that deals with Markov chains. Some sugestions are:

- L. Koralov and Y. G. Sinai,
*Theory of Probability and Random Processes* - G. Grimmett and D. Stirzaker,
*Probability and Random Processes*. (This is the textbook for Basic Probability)

- C. Gardiner,
*Stochastic Methods: A Handbook for the Natural and Social Sciences* - B. Oksendal,
*Stochastic Differential Equations* - I. Karatzas and S. E. Shreve,
*Brownian Motion and Stochastic Calculus* - L. Arnold,
*Stochastic Differential Equations: Theory and Applications* - H. Risken,
*The Fokker-Planck Equation: Methods of Solution and Applications* - P. Kloeden and E. Platen,
*Numerical Solution of Stochastic Differential Equations* - Breiman,
*Probability*

### Lecture Notes

Thanks to students and TA for finding typos/mistakes in these notes. Please let me know if you find others.

- Lecture 1: Introduction
- Lecture 2: Markov Chains (I)
- Lecture 3: Markov Chains (II)
- Lecture 4: Stochastic Processes (I)
- Lecture 5: Stochastic Processes (II)
- Lecture 6: Monte-Carlo methods
- Lecture 7: Stochastic Integration
- Lecture 8: Stochastic Differential Equations
- Lecture 9: Numerically solving SDEs
- Lecture 10: Forward and Backward Equations for SDEs
- Lecture 11: Some applications of the backward equation
- Lecture 12: Detailed balance and Eigenfunction methods
- Lecture 13: Asymptotic analysis of SDEs