Peng's Maximum Principle for Stochastic Partial Differential Equations

TL;DR

This paper extends Peng's maximum principle to more general stochastic partial differential equations with non-convex controls and control-dependent diffusion, using a novel second order adjoint state characterization.

Contribution

It introduces a new approach to characterize the second order adjoint state as a solution of a function-valued backward SPDE for broader classes of SPDE control problems.

Findings

Extended maximum principle to non-convex control domains

Characterized second order adjoint state via a backward SPDE

Applicable to general cost functionals with Nemytskii coefficients

Abstract

We extend Peng's maximum principle for semilinear stochastic partial differential equations (SPDEs) in one space-dimension with non-convex control domains and control-dependent diffusion coefficients to the case of general cost functionals with Nemytskii-type coefficients. Our analysis is based on a new approach to the characterization of the second order adjoint state as the solution of a function-valued backward SPDE.