# What is stochastic calculus

## ACM 217: Stochastic calculus and stochastic control (Spring 2007)

**Instructor:** Ramon van Handel (W. Bridge 259), ramon AT its.caltech.edu

**TA:** Yaniv Plan (Firestone 212), plan AT acm.caltech.edu

**Lectures:** Tuesday, Thursday from 10:30-12:00 a.m. (Firestone 308).

**TA office hours:** Wednesday from 10:30-11:30 a.m. (Firestone 212).

**Instr. office hours:** By appointment; email me or drop by at W. Bridge 259.

**Description:**

In the first part of this course, we will introduce the basic ideas and methods of stochastic calculus and stochastic differential equations (SDE). These techniques allow us to define rigorously the notion of a differential equation driven by white noise, and provide machinery to manipulate such equations. SDE models have a wide range of applications in many areas of science and engineering.

In the second part of the course, we will concentrate on applying these methods to stochastic control theory. Possible topics (depending on the available time and the interests of the participants in the class) include optimal stochastic control with complete observations, linear and nonlinear filtering theory, optimal stochastic control with partial observations, optimal stopping, impulse controls, stochastic stability/stabilization, and applications in science, engineering, finance, and statistics.

For a detailed overview of what this course is supposed to be about (regardless of whether we will have time to cover all the topics), see the introductory chapter in the lecture notes below.

**Prerequisites:** Introductory probability at the level of ACM 116/216. Some familiarity with elementary analysis

is helpful.

*[It has come to my attention that martingales were not covered this year in either ACM 116 or 216. As this is by far one of the most important topics in modern probability, and as we will need it desparately, we will spend some additional time in the beginning of the course going over this theory. We will probably have to give up some of stochastic control theory at the end of the course as a result.]*

**Grading:** There will be homeworks once every 2 weeks. The lowest homework grade will be dropped. There will be no midterm or final exam.

**Homeworks:**These are due on Thursdays before class (10:30 a.m.). You may hand in your homework in class, or put it in Yaniv's mailbox in the lobby of Firestone prior to class time.

- Homework 1
- Homework 2
- Homework 3
- Homework 4
- Homework 5

**Course materials:**The complete lecture notes for the course are now posted below (chapters 1-8). Suggested references for additional reading can be found in the notes.

- Lecture notes (chapters 1-8)

**Handouts:**I may distribute additional handouts either following questions in class, or when I discover omissions in the lecture notes (rather than having you print out various versions of the lecture notes).

- Handout 04/03/07. random variables with infinite moments
- Handout 04/05/07. completeness of Lp

Category: Forex

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