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.
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Dettweiler, E. On the martingale problem for Banach space valued stochastic differential equations. J Theor Probab 2, 159–191 (1989). https://doi.org/10.1007/BF01053408
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DOI: https://doi.org/10.1007/BF01053408