On almost surely periodic and almost periodic solutions of backward SPDEs

Abstract: We study linear backward stochastic partial differential equations of parabolic type with special boundary conditions in time. The standard Cauchy condition at the terminal time is replaced by a condition that holds almost surely and mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. Uniqueness, solvability and regularity results for the solutions are obtained. In particular, conditions of existence of periodic in time and "almost periodic" solutions are obtained for backward SPDEs.