Title: On the relaxed mean-field stochastic control problem,
Abstract: This paper is concerned with optimal control problems for systems governed by
mean-field stochastic differential equation, in which the control enters both
the drift and the diffusion coefficient. We prove that the relaxed state
process, associated with measure valued controls, is governed by an orthogonal
martingale measure rather that a Brownian motion. In particular, we show by a
counter example that replacing the drift and diffusion coefficient by their
relaxed counterparts does not define a true relaxed control problem. We
establish the existence of an optimal relaxed control, which can be
approximated by a sequence of strict controls. Moreover under some convexity
conditions, we show that the optimal control is realized by a strict control.