Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by L\'evy Processes

TL;DR

This paper develops a theoretical framework for fractional backward stochastic partial differential equations (BSPDEs) with applications to stochastic optimal control of systems driven by Lévy processes, including existence, uniqueness, and regularity results.

Contribution

It introduces explicit solutions for fractional BSPDEs with space-invariant coefficients and establishes well-posedness and estimates for more general equations with space-time dependent coefficients.

Findings

Explicit solution construction for fractional BSPDEs with invariant coefficients

Proved existence and uniqueness of strong solutions

Applied results to stochastic control of systems driven by Lévy processes

Abstract

In this paper, we study the Cauchy problem for backward stochastic partial differential equations (BSPDEs) involving fractional Laplacian operator. Firstly, by employing the martingale representation theorem and the fractional heat kernel, we construct an explicit form of the solution for fractional BSPDEs with space invariant coefficients, thereby demonstrating the existence and uniqueness of strong solution. Then utilizing the freezing coefficients method as well as the continuation method, we establish H\"older estimates and well-posedness for general fractional BSPDEs with coefficients dependent on space-time variables. As an application, we use the fractional adjoint BSPDEs to investigate stochastic optimal control of the partially observed systems driven by $\alpha$-stable L\'evy processes.