Information
Mondays, 1:25-3:15pm, Warren Weaver Hall (Courant Institute) 312
Instructor: Miranda Holmes-Cerfon
Office Hours: Tuesdays 4-4:50, Fridays 10:30-11:30, WWH 1107
Prerequisites: basic knowledge (e.g. undergraduate) of: probability, linear algebra, ODEs, PDEs, analysis.
Homework: Due Mondays. Download from NYU Classes website.
Grading will be based on weekly homework assignments, and (probably) a final exam. The exam will not be weighted very heavily, e.g. it may be given the weight of two homework assignments.
This is a graduate class 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, 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.
References
There is no single textbook that we will follow for the course. Lecture notes will be uploaded to the NYU Classes website for this course, as well as scans of relevant material from other sources. Here are some suggestions for books that provide additional perspective; many will be placed on reserve at the Courant library.
Books with a mathematical approach:- L. Arnold, Stochastic Differential Equations: Theory and Applications
- B. Oksendal, Stochastic Differential Equations
- L. Koralov and Y. G. Sinai, Theory of Probability and Random Processes
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus
- P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations
- L. Rogers and D. Willams, Diffusions, Markov Processes, and Martingales (Vols 1,2)
- G. Grimmett and D. Stirzaker, Probability and Random Processes
- C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences
- H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications
Weekly schedule (tentative)
Here is an outline of the topics we will cover, week-by-week. Actual topics and schedule will probably vary.
| week | day | topics |
|---|---|---|
| 1 | Jan 27 | Random variables. discrete, continuous, independence, conditioning, convergence, characteristic functions, generating functions, Borel-Cantelli Lemma |
| 2 | Feb 3 | Limit Theorems. Law(s) of Large Numbers, Central Limit Theorem, Cramer's Theorem for Large Deviations |
| 3 | Feb 10 | Markov Chains. discrete-time, continuous-time, stochastic matrix, spectral theory, ergodic theory, detailed balance |
| Feb 17 | no class — President's Day | |
| 4 | Feb 24 | Monte-Carlo methods. generating random variables, numerical integration, importance sampling, Metropolis-Hastings algorithm, kinetic Monte-Carlo |
| 5 | Mar 3 | Stochastic processes. Kolmogorov extension theorem, Markov processes, Chapman-Kolmogorov equation, generator, Gaussian processes, Karhunen-Loeve expansion, stationary processes, Wiener-Khintchine theorem, Wiener process, numerical simulation |
| 6 | Mar 10 | Stochastic processes, continued |
| Mar 14 | no class — spring break | |
| 7 | Mar 24 | Stochastic Integration. Ito integral / isometry / formula, Stratonovich integral |
| 8 | Mar 31 | Stochastic Differential Equations. existence and uniqueness, examples and applications, path integral, Girsanov transformation |
| 9 | Apr 7 | Fokker-Planck Equation. forward and backward equations, boundary conditions, first passage times, applications, Feynman-Kac formula, spectral theory, detailed balance, fluctuation-dissipation relations |
| 10 | Apr 14 | Fokker-Planck Equation, continued |
| 11 | Apr 21 | Numerical solution of SDEs. |
| 12 | Apr 28 | Asymptotic analysis of stochastic processes. |
| 13 | May 5 | Large Deviation Theory. |
| 14 | May 12 | Final exam |