On almost surely periodic and almost periodic solutions of backward SPDEs

TL;DR

This paper investigates backward stochastic partial differential equations with unique boundary conditions, establishing conditions for existence, uniqueness, regularity, and the presence of periodic or almost periodic solutions.

Contribution

It introduces new boundary conditions for backward SPDEs and provides comprehensive results on their solvability, regularity, and periodicity properties.

Findings

Existence conditions for periodic solutions

Uniqueness and regularity results for backward SPDEs

Criteria for almost periodic solutions

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.