Columbia  University   in the City of New York        |     New York, N.Y.  10027

DEPARTMENT OF MATHEMATICS                                                                 508  Mathematics Building
DEPARTMENT OF STATISTICS                                                                        618  Mathematics Building
                                                                                                                                                  2990 Broadway

Fall Semester 2003
Professor Ioannis Karatzas

G8263:  STOCHASTIC  DIFFERENTIAL EQUATIONS  AND  APPLICATIONS

COURSE SYLLABUS

Prerequisites:  A course on Real Analysis and Probability, at the level of G4151- 4153 or G6105-6106.

Overview of Stochastic Analysis:  Elements of the general theory of random pro-cesses. Martingales. Doob-Meyer decomposition. Square-integrable, Local- and Semi-Martingales. The Brownian Motion process, connections with partial diffe-rential equations. The Poisson problem and the stochastic representation of its solution.

Integration  with respect to continuous semi-martingales, Ito’s change-of-variable formula, Girsanov’s theorem and its applications. The martingale representation property for the Simple Random Walk and for the Brownian filtration, the Clark-Ocone formula.  Least-squares approximation of random variables by stochastic integrals. Notion and properties of Local Time.

Stochastic Differential Equations and Diffusion Processes:  The Ito-Doeblin theory of diffusions. Existence and Uniqueness results. The martingale problem of Stroock and Varadhan. Linear diffusions. Connections with partial differential equations.

Optimal Stopping and Dynamic Allocation problems.
Stochastic Control problems: the general probabilistic theory, examples.
Filtering problems: the Kalman-Bucy filter, enlargements of filtrations.
Mathematical Finance: Hedging, Portfolio Optimization, Diversity, Completeness.

Special Topics (as time permits): Study of the Brownian local time, Bessel processes, the so-called 2M-X property, Hypercontractivity, Skorohod embedding, Stochastic Differential Geometry, Excursions.

RECOMMENDED TEXT:

I.KARATZAS & S.E.SHREVE (1988)  Brownian Motion and Stochastic Calculus.  Springer-Verlag, New York.