Singular mean-field optimal control for forward-backward stochastic systems and applications to finance

Abstract

In this paper, we study a class of singular stochastic optimal control problems for systems described by mean-field forward-backward stochastic differential equations, in which the coefficient depend not only on the state process but also its marginal law of the state process through 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 control. Necessary conditions for optimal control for this systems in the form of a Pontrygin maximum principle are established by means convex perturbation techniques for both continuous and singular parts. Our stochastic maximum principle differs from the classical one in the sense that here the adjoint equation has a mean-field type. The control domain is assumed to be convex. As an illustration of our results, we consider a mean-variance portfolio selection mixed with a recursive utility functional optimization problem involving singular control. The explicit expression of the 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 with time-inconsistent solution.Keywords Singular stochastic control · Mean-field forward-backward stochastic differential equation · Mean-field type maximum principle · Time-inconsistent control problem · McKean-Vlasov systems M. Hafayed

On Mean-Field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Lévy Processes

“…We refer the readers to [13,16,22,23] for other models of conjecture. Applying Itô's formula to (38), we have…”

“…Therefore, it is natural to investigate optimal control problems for systems governed by this kind of stochastic equations. Stochastic control problems for mean-field SDEs have been studied by many authors; see, for example, [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The second-order necessary and sufficient conditions of near-optimal singular control for mean-field SDE have been established in [12].…”

“…The necessary and sufficient conditions for near optimality for mean-field jump diffusions with applications have been derived in [15]. The singular optimal control problem for mean-field FBSDEs with its applications has been investigated in [16]. A general mean-field maximum principle for SDEs has been established in [18].…”

Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem

“…The necessary and sufficient conditions for near-optimality of meanfield jump diffusions with applications have been derived by Hafayed et al [14]. Singular optimal control for mean-field forward-backward stochastic systems and applications to finance have been investigated in Hafayed [15]. Second-order necessary conditions for optimal control of mean-field jump diffusion have been obtained by Hafayed and Abbas [16].…”

Stochastic maximum principle for delayed backward doubly stochastic control systems

“…Yong [18] gave the optimality variational principle for controlled forward backward stochastic differential equations with mixed initial-terminal conditions. Hafayed et al [3][4][5] gave the maximum principle and application to finance in mean field optimal control for forward backward stochastic systems and so on.…”