Necessary condition for optimal control of doubly stochastic systems

Abstract

The aim of this paper is to establish a necessary condition for optimal stochastic controls where the systems governed by forward-backward doubly stochastic differential equations (FBDSDEs in short). The control constraints need not to be convex. This condition is described by two kinds of new adjoint processes containing two Brownian motions, corresponding to the forward and backward components and a maximum condition on the Hamiltonian. The proof of the main result is based on spike's variational principle, duality technique and delicate estimates on the state and the adjoint processes with respect to the control variable. An example is provided for illustration.

On mean-field control problems for backward doubly stochastic systems

“…Motivated by the existing works, in this paper we investigate the mean-field control problem (1.1)-(1.2) for MF-BDSDEs and aim to obtain SMP. We remark that Han et al [12] has obtained SMP for control problems involving such BDSDEs without mean-field terms (see also [29,30]). In our control problem (1.1)-(1.2), the state process and the cost functional both depend on the joint distribution of the state process and the control process.…”