On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application

Abstract

In this paper, we are concerned with an optimal control problem where the system is driven by fully coupled forward-backward stochastic differential equation of mean-field type with risk-sensitive performance functional. We study the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of the initial stochastic control problem in this type of risk-sensitive performance problem, where an admissible set of controls are convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. Finally, we illustrate our main result of this paper by giving two examples of risksensitive control problem under linear stochastic dynamics with exponential quadratic cost function, the second example will be a mean-variance portfolio with a recursive utility functional optimization problem involving optimal control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback.

A risk-sensitive global maximum principle for controlled fully coupled FBSDEs with applications

“…By assuming that the control domain is convex, necessary and sufficient maximum principles are obtained. Results for mean-field type control is soon given in Chala and Hafayed [7]. Both these results generalized the pioneer work of Wu [40] from risk-neutral case to risk-sensitive case.…”

“…The main techniques in [17] are a new first-and second-order spike variational method which was first introduced by Hu [16] and a new decoupling relation between first-and second-order variations of x(•), y(•) and z(•). For ease of use, very recently, Lin and Shi [25] extended the above result from the recursive cost functional (7) to the case with the cost functional of the following general form:…”

Stochastic maximum principle for partially observed risk‐sensitive optimal control problems of mean‐field forward‐backward stochastic differential equations

“…They also presented the optimal solution with respect to the small period time, large period time, the risk‐seeking and risk averse respectively with numerical investigation. Chala and Hafayed 47 provided the necessary and sufficient optimality conditions for fully coupled FBSDE of mean‐field type with risk‐sensitive performance. Chala 27 set up a necessary stochastic maximum principle for the risk‐sensitive optimal control for a backward stochastic system.…”

“…(b) Unlike the system with full information studied by References 45‐47,62, we consider the partially observed optimal control problem on convex domain. This extension is also not easy, since by including the noisy process Y , there is an auxiliary variational equation (see (9)) in addition to the variational equation (see (8)).…”

“…This extension is also not easy, since by including the noisy process Y , there is an auxiliary variational equation (see (9)) in addition to the variational equation (see (8)). In addition, we give some prior estimates (see (11)), which did not appear in the full information case studied by References 45‐47,62. We also introduce four adjoint equations, including three mean‐field backward SDEs and one mean‐field forward SDE, which increase the difficulty of analyzing the maximum condition and improve the skills of pure mathematical calculation.…”