On backward stochastic evolution equations in Hilbert spaces and optimal control

In this paper a new result on the existence and uniqueness of the adapted solution to a backward stochastic evolution equation in Hilbert spaces under a non-Lipschitz condition is established. The applicability of this result is then illustrated in a discussion of a concrete backward stochastic partial differential equation. Furthermore, a stochastic maximum principle for optimal control problems of stochastic systems governed by backward stochastic evolution equations in Hilbert spaces is obtained.