On the martingale problem for Banach space valued stochastic differential equations

Abstract

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet≥0 and the continuous functionsf:[0,t] →E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ∫ 0 . Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ∫ 0 . Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY =∫ 0 . Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.