# Stochastic control of mean-field SPDEs with jumps

## TL;DR

This paper develops a maximum principle for controlling mean-field stochastic PDEs with jumps under partial information, introducing general mean-field operators and applying results to an optimal harvesting problem.

## Contribution

It introduces a novel maximum principle for mean-field SPDEs with jumps, incorporating general mean-field operators and partial information, with proven existence and uniqueness results.

## Key findings

- Established a maximum principle for the control problem.
- Proved existence and uniqueness of solutions for the involved equations.
- Derived explicit optimal control for an harvesting application.

## Abstract

We study the problem of optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in the case of \textit{partial information} control. One important novelty of our problem is represented by the introduction of \textit{general mean-field} operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We finally apply our results to find the explicit optimal control for an optimal harvesting problem.