Optimal Partial Information Control of SPDEs with Delay and Time-Advanced Backward SPDEs

Abstract

We study optimal control problems for (time-) delayed stochastic partial differential equations (SPDEs) with jumps and with partial information flow available to the controller. We establish sufficient and necessary (Pontryagin/Bismut/Pardoux-Peng type) maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-) advanced backward stochastic partial differential equation (ABSPDE). Several results on existence and uniqueness of such ABSPDEs are shown. The results are illustrated by an application to a harvesting problem from a biological system (e.g. a fish population), where the dynamics of the population is modeled by a stochastic reaction-diffusion equation. IN N OVATION S IN S TOCHAS TIC AN AL Y S IS AN D AP P L ICAT ION S with emphasis on S T OC H A S T I C C ON T R OL A N D I N F OR M A T I ON SECTION 1: INTRODUCTION Partial information optimal harvesting of a system with delay Let B(t) = B(t, ω) be a Brownian motion and N(dt, dζ) := N(dt, dζ)− ν(dζ)dt, where ν is the Levy measure of the jump measure N(·, ·), be an independent compensated Poisson random measure on a filtered probability space (Ω,F , {Ft}0≤t≤T ,P). IN N OVATION S IN S TOCHAS TIC AN AL Y S IS AN D AP P L ICAT ION S with emphasis on S T OC H A S T I C C ON T R OL A N D I N F OR M A T I ON If a fish population of density Y (t, x) at time t and at the point x is exposed to a harvesting rate density c(t, x) ≥ 0, the corresponding population state dynamics may be modeled by the following equation: dY (t, x) = ( 1 2 ∆Y (t, x) + αY (t, x) + βY (t − δ, x)− c(t, x))dt + σ0Y (t, x)dB(t) + ∫ R γ0(ζ)Y (t −, x)N(dt, dζ), (1.1) where ∆ = n ∑ i=1 ∂2 ∂x2 i is the Laplacian operator acting on x . Here α, β, σ0 are constants and γ0 is deterministic. IN N OVATION S IN S TOCHAS TIC AN AL Y S IS AN D AP P L ICAT ION S with emphasis on S T OC H A S T I C C ON T R OL A N D I N F OR M A T I ON This is a stochastic partial differential equation (SPDE) of reaction-diffusion type. The Laplacian operator models the diffusion (distribution in space), while the other terms model the local growth at each point x . For biological reasons it is natural to include a delay term like βY (t − δ, x) in the dynamics. IN N OVATION S IN S TOCHAS TIC AN AL Y S IS AN D AP P L ICAT ION S with emphasis on S T OC H A S T I C C ON T R OL A N D I N F OR M A T I ON Suppose we want to find a harvesting rate density c(t, x) which maximizes the total expected utility of the harvest plus the utility of the remaining population at a terminal time T > 0. We assume that at any time t the controller (harvester) has only a partial information flow Et available to base her decision on. Then the problem is to maximize