Near-optimality conditions in mean-field control models involving continuous and impulse controls

Abstract

In this paper we discuss stochastic control models which are described by a stochastic differential equation of mean-field type, in the sense that the coefficients are permitted to depend on the state process as well as of its expected value. The control variable has two components, the first being absolutely continuous and the second is a piecewise impulse process which is not necessarily increasing. Necessary and sufficient conditions for a control to be near optimal are studied in the form of stochastic maximum principle by using Ekeland’s variational principle, which allows to produce two approximate variational inequalities in integral form. The first inequality is constructed by the spike variation technique in terms of the H -function employed for absolutely continuous part of all near optimal control. The second one is defined in term of the first order adjoint process by using a convex perturbation technique for all near optimal impulse controls.