Stochastic Differential Equations: Theory and Applications

Course Announcement Spring 2014

Stochastic Differential Equations: Theory and Applications (Listed as: Math 5415 under ‘Probabilty and Martingales’)

Nathan Glatt-Holtz [email protected] http://www.math.vt.edu/people/negh/

Stochastic differential equations (SDEs) play an important role across the physical, biological and social sciences with applications ranging from population biology to turbulence in fluids and of course in the pricing of financial instruments. These equations and their surrounding theory are also of intrinsic mathematical interest representing a rich class of stochastic processes with connections to numerous other areas of probability theory and partial differential equations. This course will serve as an introduction to both the theoretical and applied aspects of the subject. To begin we will introduce some basic foundations: Brownian motion, Martingale theory, the It¯o-Doeblin stochastic integral, the It¯o formula and the basics of the stochastic calculus. We will then use these tools for the study of stochastic differential equations both in their Markovian and pathwise frameworks. As time permits and depending on the interests of the students additional topics may include: numerical schemes for the simulation of SDEs, parameter estimation and filtering, further connections with and applications to partial differential equations. Prerequisites for participation in the course will be minimal: an advanced undergraduate level background in probability theory and differential equations should be sufficient. A strong preparation in analysis equivalent to a graduate course involving measure theory will be necessary to follow the details of all the arguments. However, I will taylor the presentation to allow students with less background to follow the more formal aspects of the theory. The main reference text will be [Øks03] but other good general references includes [Eva13, Arn74, Fri06, KS91, KP92]. Course assessment will be based on assigned problems sets or a presentation as suits the background of the students.

References [Arn74] L. Arnold, Stochastic differential equations: theory and applications [Eva13] L. C. Evans, An introduction to stochastic differential equations. [Fri06] Avner Friedman, Stochastic differential equations and applications. [KP92] P. Kloeden and E. Platen, Numerical solution of stochastic differential equations. [KS91] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus. [Øks03] B. Øksendal, Stochastic differential equations.