On forward and backward SPDEs with non-local boundary conditions

TL;DR

This paper investigates linear parabolic stochastic PDEs with non-local boundary conditions that involve values of the solution at multiple times, including the initial, terminal, and continuously distributed times, with applications to portfolio selection.

Contribution

It introduces a novel class of boundary conditions for SPDEs that combine values at different times, establishing existence, uniqueness, and regularity results for these equations.

Findings

Established solvability and regularity for non-local boundary SPDEs.

Extended the framework to cover almost surely periodic backward equations.

Discussed potential applications to portfolio selection problems.

Abstract

We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.