Bernt Øksendal
Singular control and optimal stopping of
memory mean-field processes
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 singular stochastic control and optimal stopping problems of memory mean-field processes.
We consider the space M of measures on R with norm introduced by Agram and Ø. , and prove the existence and uniqueness of solutions of reflected memory mean-field backward stochastic differential equations.
We prove two stochastic maximum principles for singular control of such systems, 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 variational inequalities, one of them with values in the space of bounded linear functionals on path segment spaces. Then we investigate the relation to optimal stopping of such processes. The talk is based on recent joint work with Nacira Agram, Achref Bachouch and Frank Proske, all at the University of Oslo.