Dr. Dmitry Kramkov (Carnegie Mellon)

Title:Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

Abstract:Let Q and P be equivalent probability measures and let psi be a J-dimensional vector of random variables such that d Q/ d P and psi are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale S_t \set E^{Q}[psi|F_t]. While the drift b=b(t,x) and the volatility sigma = sigma(t,x) coefficients for X need to have only minimal regularity properties with respect to x, they are assumed to be analytic functions with respect to t. We provide a counter-example showing that this $t$-analyticity assumption for sigma cannot be removed. 

The presentation is based on a joint work with Silviu Predoiu; see http://arxiv.org/abs/1110.3248