# Singular control of SPDEs with space-mean dynamics

## TL;DR

This paper develops maximum principles for optimal singular control of SPDEs with space-mean dependence, modeling population growth in random environments, and introduces a reflected BSPDE framework with existence and uniqueness results.

## Contribution

It introduces a novel control framework for SPDEs with space-mean dependence, including maximum principles and a new class of reflected BSPDEs with proven well-posedness.

## Key findings

- Derived necessary and sufficient maximum principles for control.
- Established existence and uniqueness of the reflected BSPDEs.
- Applied the theory to optimal population harvesting models.

## Abstract

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.