### Information

Mondays, 1:25-3:15pm, Warren Weaver Hall 512

Instructor: Miranda Holmes-Cerfon

Office Hours: (tentatively) Thursdays 4-6pm, WWH 1107

**Prerequisites:**
Basic Probability (or equivalent masters-level probability course), and good upper level undergraduate or beginning graduate knowledge of linear algebra, ODEs, PDEs, and analysis.

**Description:**
This course 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, numerical algorithms. 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. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

The course will be divided roughly into two parts, taking roughly an equal amount of time: Part I will focus on stochastic processes, and Part II will focus on stochastic calculus.

Homework will be a critical part of the course. Lectures will mostly be theory, and examples or extensions will be assigned as homework problems. You must do these if you want to learn something from the course. Homework will require some computing, preferably in Python or Matlab. Students without programming experience will have to put in extra effort in the first few weeks.

### References

There are three textbooks that are not required, but that are highly recommended:- G. A. Pavliotis,
*Stochastic Processes and Applications*. - 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*.

Other good references include:

- B. Oksendal,
*Stochastic Differential Equations* - L. Koralov and Y. G. Sinai,
*Theory of Probability and Random Processes* - R. Durrett,
*Essentials of Stochastic Processes* - R. Durrett,
*Stochastic Calculus: A Practical Introduction* - I. Karatzas and S. E. Shreve,
*Brownian Motion and Stochastic Calculus* - L. Arnold,
*Stochastic Differential Equations: Theory and Applications* - P. Kloeden and E. Platen,
*Numerical Solution of Stochastic Differential Equations* - Breiman,
*Probability*

### Lecture Notes

Thanks to the students of ASA 2017 (and students of previous years) for finding typos/mistakes in these notes. These notes are continuously evolving, so please let me know if you find other mistakes in them.

- Syllabus
- Lecture 1: Introduction
- Lecture 2: Markov Chains (I)
- Lecture 3: Markov Chains (II): Detailed balance and Markov Chain Monte Carlo
- Lecture 4: Continuous-time Markov chains
- Lecture 5: Stochastic Processes I: Introduction
- Lecture 6: Stochastic Processes II: Spectral decomposition for sstationary processes
- Lecture 7: Brownian motion
- Lecture 8: Stochastic Integration
- Lecture 9: Stochastic differential equations
- Lecture 10: Numerically solving SDEs
- Lecture 11: Forward and backward equations for SDEs
- Lecture 12: Detailed balance and Eigenfunction methods
- Lecture 13: Some applications of the backward equations

In addition to the lectures: - Asymptotic analysis of SDEs (Lecture 13 of ASA 2015)