Maximum Principles for Partially Observed Controls of Forward SPDEs and Backward SDEs with Jumps

TL;DR

This paper develops maximum principles for partially observed control problems involving coupled forward SPDEs and backward SDEs with jumps, introducing new methods for well-posedness and stochastic flow representations.

Contribution

It introduces two versions of Pontryagin maximum principles for systems with jumps, including novel well-posedness results for operator-valued SPDEs with jumps.

Findings

Established maximum principles for controlled FSPDE-BSDE systems with jumps.

Proved well-posedness of singular backward SPDEs with jumps.

Developed a Malliavin calculus approach for systems with random coefficients.

Abstract

This work establishes two versions of the Pontryagin-type maximum principles for partially observed optimal control of coupled forward stochastic partial differential equations (FSPDEs) and backward stochastic differential equations (BSDEs) with jumps in convex control domains. The FSPDE-BSDE system is driven by cylindrical Wiener processes, finite-dimensional Brownian motions, and compensated Poisson random measures. For systems with deterministic coefficients, a direct method is employed and particular attention is focused on establishing the well-posedness of a singular backward SPDE with jumps. For systems with random coefficients, a Malliavin calculus approach is developed. The main novelty here is the establishment of the well-posedness of an operator-valued SPDE with jumps, which provides a new stochastic flow representation for linear SPDEs with jumps.