# Stochastic Calculus

## MATH-GA 2902.001

### Courant Institute of Mathematical Sciences, New York University Fall Semester, 2018 Lectures: Monday, 7:10 to 9 pm, Room 109, Warren Weaver Hall

#### Instructor: Jonathan Goodman, his web page, email: goodman@cims.nyu.edu phone: 212-998-3326, office: 529 Warren Weaver Hall office hours: 4 to 6 pm Wednesday or by appointment (call or email for a time)

##### Course description

This is a course on stochastic processes intended for people who will apply these ideas to practical problems. It covers mathematical terminology used to describe stochastic processes, including filtrations and transition probabilities. It uses some measure theoretic terminology but is not mathematically rigorous. The emphasis is on analytical tools (forward, backward equations, etc.) and computational methods (difference equations, simuation, Monte Carlo) for studying specific processes.

We start with Brownian motion and diffusion processes described by their short time mean (drift) and variance (quadratic variation). These are related to backward and forward partial differential equations. We discuss how to develop a diffusion process to model a physical or financial process. The Ito calculus is developed and related to stochastic differential equations (SDEs), change of measure (Girsanov) and the Feynman Kac formula. Linear Gaussian processes are described in more detail, in continuous and discrete time, with applications to filtering (estimation from noisy measurements). Time permitting, we may describe discrete Markov chains and give some analogous theory for them.

Prerequisites: A mastery of multivariate calculus, multivariate probability, and linear algebra is required. There is an assignment 0 (see the assignments page) to check that you are ready for the class. Assignments will require some lightweight computing in Python 3. Previous experience with Python specifically is not needed, but students should have some exposure to programming in R, or C, or C++, Java, Python, Matlab, VBA, Fortran, etc.

Assignments, exams, grading: The final grade will be based on weekly homework assignments and a final exam. Assignments will be given each week. These are due in class, in paper, at the beginning of the following class.

Communication: Please use the Forum page of the NYU Classes site for this course for all content related communication, including questions about assignments, lectures, or notes. Feel free to contact the instructor or TA directly about other issues such as appointments, missed classes, late assignments, grading issues, etc. The instructor and TA will check the message board frequently. Look there for important course announcements, in particular corrections to assignments.