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

Abstract

This paper deals with a partially observed risk‐sensitive optimal control problem described by mean‐field forward‐backward stochastic differential equations and the cost functional is a mean‐field exponential of integral type. By using Girsanov's theorem as well as classical convex variational techniques, we obtain two risk‐sensitive maximum principles, which are characterized in terms of the variational inequalities. Moreover, under certain concavity assumption, we give the sufficient condition for the optimality. Through the application of results, we consider the linear‐quadratic risk‐sensitive optimal control problem under partially observed information and fully observed information, respectively.

Robust risk‐sensitive control

“…We now focus on the corresponding Lyapunov differential Equation ( 24) and the Riccati differential Equations ( 25) and (26) and show that the sufficient conditions of Theorem 3 hold. For the matrix A * in (23), which in this example is a scalar, we have:…”

“…The risk‐sensitive control problem for linear stochastic systems with additive noise was introduced by Jacobson 13 who found an explicit closed‐form solution in a linear state‐feedback in the case of full observations. For risk‐sensitive control with partial observations see, for example, References 14–16, for discrete‐time systems see, for example, References 17,18, for connections with robust control see, for example, References 16,19–21, for the risk‐sensitive maximum principle see, for example, References 22–24, for the risk‐sensitive control of mean‐filed systems see, for example, References 24–26 and 27, for the Hamilton‐Jacobi‐Bellman equation of risk‐sensitive control see Reference 28, for the risk‐sensitive differential games see, for example, References 29–35, and for more general exponential criteria that admit explicit closed‐form solutions see References 36–39. The risk‐sensitive control is particularly suitable for optimal investment problems, see, for example, References 36,38–43.…”

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

“…A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the BSDE and the presence of jump diffusions. Previously, for such problems, only stochastic maximum principles have been established, which in general constitute only the necessary conditions for optimality in terms of variational inequalities (see References 34,39‐51 and the references therein). The notable result of the stochastic maximum principle for control of FBSDEs was obtained in Reference 41.…”

“…However, the problems in Reference 43,46 are simplified in that their BSDEs are defined as objective functionals, and their FBSDEs are restricted to one‐dimensional processes. We also mention that the work 39 considered the stochastic control for fully coupled mean‐field type FBSDEs with jump diffusions, and the work 50 studies the partially observed risk‐sensitive control for FBSDEs. Moreover, the work 49 considered the stochastic control problem for mixed FBSDEs, where the “mixed” means that there are deterministic and random controllers.…”

“…Moreover, the work 49 considered the stochastic control problem for mixed FBSDEs, where the “mixed” means that there are deterministic and random controllers. The approaches in References 39,49,50 are also closely related to the stochastic maximum principle framework. However, the control spaces in References 39,49,50 are restricted to be convex sets, and the monotonicity condition is imposed in Reference 39 for the corresponding FBSDE.…”