Abstract
This paper studies first a result of existence and uniqueness of the solution to a backward stochastic differential equation driven by an infinite-dimensional martingale. Then, we apply this result to find a unique solution to a backward stochastic partial differential equation in infinite dimensions. The filtration considered is an arbitrary right-continuous filtration, not necessarily the natural filtration of a Wiener process. This, in particular, allows us to study more applications, for example, the maximum principle for a controlled stochastic evolution system. Some examples are discussed in the paper as well.