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September  2020, 9(3): 817-843. doi: 10.3934/eect.2020035

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

 Laboratory of Applied Mathematics, University Mohamed Khider, P.O. Box 145, Biskra 07000. Algeria

Received  October 2018 Revised  October 2019 Published  March 2020

Fund Project: The first author is supported by PRFU project N: C00L03UN070120180002

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 risk-sensitive 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.

Citation: Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
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