Mean-field Risk-sensitive Stochastic Control Problems With Jumps Under Partial Information

In this paper we mainly study mean-field risk-sensitive stochastic control problems with jumps under partial information.Our research is divided into two parts.In the first part of this paper(Chapter 3),we study risk-sensitive optimal control problems of mean-field forward backward stochastic differential equations with jumps under partial information when the control domain is convex.We give the necessary condition of risk-sensitive optimal control which is known as Pontryagin’s stochastic maximum principle and the sufficient condition of risk-sensitive optimal control problem.To be more precise,the stochastic control system with jumps is as follows:Since state variables x(·),y(·),z(·)cannot be completely observed,we have to use the noise process y(·)to make x(·),y(·),z(·)be partially observed,where stochastic process Y(·)satisfies the following equation:The cost functional is defined in the following:(?) where θ is the risk-sensitive coefficient.The control ū(·)which maximizes the cost functional is optimal.Risk-sensitive optimal control problem:To find the optimal control ū(·)∈Uad satisfyingOur aim is to study the conditions satisfied by the optimal control ū(·)of the risksensitive optimal control problem.we obtain the maximum principle of risk-sensitive optimal control problem by using Girsanov theorem and classical convex variational method.Furthemore,the sufficient conditions of optimal control are given under certain concavity assumptions.Its application in the linear quadratic optimal control problem is illustrated with an example.In the second part of the paper(Chapter 4),we study the risk-sensitive optimal control problem of mean-field stochastic differential equations with jumps under partial information when the control domain is not necessarily convex.We mainly study the necessary condition.of risk-sensitive optimal control,i.e.,Peng’s stochastic maximum principle.To be more precise,the stochastie control system with jumps is as follows:Since state variables x(·)cannot be completely observed,we still have to use the noise process Y(·)to make x(·)be partially observed,where stochastic process Y(·)satisfies the following equation:The cost functional is defined in the following:Risk-sensitive optimal control problem:To find the optimal control ū(·)∈ Uad satisfyingOur aim is to study the conditions satisfied by the optimal control ū(·)of the risksensitive optimal control problem.Since the control domain is not always convex,the convex variational method mentioned above cannot be used here.We have to use the spike variational method and Girsanov theorem to obtain the maximum principle of risksensitive optimal control problem.In the end,we give an example to explain its application in the linear quadratic optimal control problem.