Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 236B: Stochastic Dynamics and Applications
Approved: 2009-03-01, Alfred Fannjiang

Units/Lecture:

Offered irregularly; 4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Stochastic Calculus for Finance II by S. E. Shreve, ($70)
Search by ISBN on Amazon: 0387401016

Prerequisites:

MAT 236A, or consent of instructor.

Course Description:

Stochastic processes, Brownian motion, Stochastic integration, martingales, stochastic differential equations. Diffusions, connections with partial differential equations, mathematical finance.

Suggested Schedule:

Department Syllabus
MAT 236B: Stochastic Dynamics and Applications

Lectures Sections Topics/Comments
3 Chapters 2 - 3 Review: Information/conditioning, Brownian motion, first passage time
2 4.5 Black-Scholes-Merton equation
2 5.1 - 5.2 Arbitrage theorem, risk-neutral measure, Black-Scholes-Merton formula
3 5.3 - 5.4 Fundamental theorems in asset pricing
1 5.5 Dividend paying stocks
1 5.6 Forwards and futures
3 Chapter 7 Exotic options: barrier options, lookback options, Asian options
3 Chapter 8 American derivatives
3 Chapter 9 Change of numeraire: foreign and domestic risk-neutral measure, forward measure
3 Chapter 10 Term-structure models
5 Chapter 11 Rare events in finance, jump-diffusion processes

Additional Notes:

COMMENT: This book focuses on the financial application of stochastic calculus. I have taught similar topics from other sources in 2006-2007 but did not find an ideal textbook that would give clear, in-depth coverage on both stochastic and finance. I hope Shreve's book will fit the bill (I am teaching from it this year, 2008-2009). For physical applications, the book Handbook of Stochastic Methods by Gardiner, comes to mind. It covers many interesting topics in physics and its best part is on the analysis/application of Fokker-Planck and master equations (rather than the stochastic calculus).