Existence of weak solutions for stochastic differential equations with driving semimartingales

Abstract

We consider the following stochastic differential equation , Where Z is a given finite-dimensional semimartingale and K is given predictable process with right-continuous and left-hand limited paths, on some filtered probability space . The coefficient gis assumed to be bounded (an assumption that can be slightly relaxed), and may depend on the whole path of X, in a predictable way. We prove that this equation admits at least one week solution whenever gs(ω,.) is continous on the Skorokhod space (d=dimension of X) endowed with the uniform topology

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