Stochastic Control of Memory Mean-Field Processes

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 Øksendal (Model uncertainty stochastic mean-field control. arXiv:1611.01385v5, [2]), 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 (time-advanced backward stochastic differential equations (absdes), 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.