# Stochastic Control of Memory Mean-Field Processes

## TL;DR

This paper develops stochastic control methods for memory mean-field processes, establishing existence, uniqueness, and maximum principles, and applies these to solve specific control problems involving memory effects.

## Contribution

It introduces new stochastic maximum principles for memory mean-field processes and proves existence and uniqueness of solutions for related stochastic differential equations.

## Key findings

- Established existence and uniqueness of solutions for memory mean-field stochastic equations.
- Proved two stochastic maximum principles under partial information.
- Solved a memory mean-variance and a linear-quadratic control problem.

## Abstract

By a memory mean-field process we mean the solution $X(\cdot)$ of a stochastic mean-field equation involving not just the current state $X(t)$ and its law $\mathcal{L}(X(t))$ at time $t$, but also the state values $X(s)$ and its law $\mathcal{L}(X(s))$ at some previous times $s<t$. Our purpose is to study stochastic control problems of memory mean-field processes. - We consider the space $\mathcal{M}$ of measures on $\mathbb{R}$ with the norm $|| \cdot||_{\mathcal{M}}$ introduced by Agram and {\O}ksendal in \cite{AO1}, and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. - We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of \emph{(time-) advanced backward stochastic differential equations}, one of them with values in the space of bounded linear functionals on path segment spaces. - As an application of our methods, we solve a memory mean-variance problem as well as a linear-quadratic problem of a memory process.