By a memory mean-field process we mean the solution X(⋅) of a
stochastic mean-field equation involving not just the current state X(t) and
its law L(X(t)) at time t, but also the state values X(s) and
its law 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 M of measures on R with the
norm ∣∣⋅∣∣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