Stochastic Control of Memory Mean-Field Processes

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

Mean-Field Stochastic Control with Elephant Memory in Finite and Infinite Time Horizon

Our purpose of this paper is to study stochastic control problem for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. - In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. - For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system