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We consider mean-field type stochastic optimal control problems for systems governed by special mean-field forward-backward stochastic differential equations with jump processes, in which the coefficients contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. Necessary conditions of optimal control for these systems in the form of a maximum principle are established by means of spike variation techniques. Our result differs from the classical one in the sense that here the adjoint equation has a meanfield type. The control domain is not assumed to be convex.

This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type (MFSDEs in short), in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular. We establish necessary as well as sufficient conditions for optimal singular stochastic control where the system evolves according to MFSDEs. These conditions of optimality differs from the classical one in the sense that here the adjoint equation turns out to be a linear mean-field backward stochastic differential equation. The proof of our result is based on convex perturbation method of a given optimal control. The control domain is assumed to be convex. A linear quadratic stochastic optimal control problem of mean-field type is discussed as an illustrated example. Keywords Stochastic optimal singular control • Mean-field stochastic maximum principle • Mean-field necessary and sufficient conditions of optimality • McKean-Vlasov SDEs • Convex perturbation

In this paper, we study mean-field type stochastic control problems for systems described by mean-field stochastic differential equations with jump processes, in which the coefficients contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. We derive necessary as well as sufficient conditions of near-optimality for our model, using Ekeland's variational principle, spike variation method and some estimates of the state and adjoint processes. Under certain concavity conditions with non-negative derivatives, we prove that the near-maximum condition on the Hamiltonian function in integral form is a sufficient condition for nearoptimality. Our result differs from the classical one in the sense that here the adjoint equation has a mean-field type, while the second-order adjoint equation remains the same as in the classical case. As an application, our results are applied to a mean-variance portfolio selection where explicit expression of the near-optimal portfolio selection strategy is obtained in the state feedback form involving both state process and its marginal distribution, via the solutions of Riccati ordinary differential equations.

In this paper, we study the mean-field-type partial information stochastic optimal control problem, where the system is governed by a controlled stochastic differential equation, driven by the Teugels martingales associated with some Lévy processes and an independent Brownian motion. We derive necessary and sufficient conditions of the optimal control for these mean-field models in the form of a maximum principle. The control domain is assumed to be convex. As an application, the partial information linear quadratic control problem of the mean-field type is discussed.