On the Martingale Problem for Infinite Dimensional Stochastic Differential Equations.

Abstract

The purpose of this dissertation is to establish, in a certain infinite dimensional setting, some existence and uniqueness results pertaining to the type of exponential martingale problem previously considered by Strook and Varadhan in the finite dimensional setting. The setting is among those considered by K. Ito in his work on stochastic differential equations in infinite dimensional spaces. Briefly, the martingale problem takes the following form: to show the existence and uniqueness of a measure m on the space S of continuous functions defined on the nonnegative real lien with values in some state space, having the property that exp(f(t,w)) is an m-martingale, f being a function of type, with t denoting a time parameter, and w denoting a generic element of S. The measure m is then called a solution of the martingale problem. In this dissertation, the state spaces under consideration are subspaces of the space of tempered distributions on the real line. The existence of solutions is established under certain continuity conditions by constructing a family of measures satisfying a Prokhorov criterion for relative compactness with respect to weak convergence, and then showing that a weak limit point of that family has the desired martingale property. Secondly, under somewhat stronger assumptions, a uniqueness result for a previously constructed solution is proved in a manner similar to that in which the uniqueness of a related stochastic differential equation is established. Finally, the relation of the martingale problem to certain infinite dimensional stochastic integrals and stochastic equations is pointed out.