Singular mean-field control games with applications to optimal harvesting and investment problems

This paper studies singular mean field control problems and singular mean field stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Applications are given to optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and to mean-field singular investment games. In particular, a simple singular mean-field investment game is studied where the Nash equilibrium exists but is not unique

Maximum principle for discrete time mean-field stochastic optimal control problems

In this paper, we study the optimal control of a discrete-time stochastic differential equation (SDE) of mean-field type, where the coefficients can depend on both a function of the law and the state of the process. We establish a new version of the maximum principle for discrete-time stochastic optimal control problems. Moreover, the cost functional is also of the mean-field type. This maximum principle differs from the classical principle since we introduce new discrete-time backward (matrix) stochastic equations. Based on the discrete-time backward stochastic equations where the adjoint equations turn out to be discrete backward SDEs with mean field, we obtain necessary first-order and sufficient optimality conditions for the stochastic discrete optimal control problem. To verify, we apply the result to production and consumption choice optimization problem

Stochastic Maximum Principle for the System Governed by Backward Doubly Stochastic Differential Equations with Risk-Sensitive Control Problem and Applications

his thesis based on the study of the stochastic maximum principle with risk-sensitive for two different systems. We obtain these systems by generalizing the results of Chala [10; 11], and by using the paper of Djehiche et al. in [13]: The first system is driven by a backward doubly stochastic differential equation. We use the risk-neutral model for which an optimal solution exists as a preliminary step, this is an extension of the initial control problem. Our goal is to establish necessary and sufficient optimality conditions for the risk-sensitive performance functional control problem. We show for the second system which is driven by a fully coupled forward-backward stochastic differential equation of mean-field type, by using the same technique as in the first case, we get the necessary and sufficient optimality conditions for the risk-sensitive, where the set of admissible controls is convex in all the cases. Finally, we illustrate our main results by giving applied examples of risk-sensitive control problems

Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results