An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

Abstract

In this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

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

“…The maximum principle of risk-neutral obtained by Min et al [18], is similar to ours (Theorem 3.1), but the adjoint equations and maximum conditions depend heavily on the risk-sensitive parameter. If we put θ = λ = µ = κ = 0, we can compare our feedback control of (66) with the control obtained by Hafayed et al [12].…”

“…for given functions Φ, Ψ and l. This cost functional is also of mean-field type and with exponential expected, as the functions Φ, Ψ and l depend on the marginal law of the state process through its expected value. The pioneering works on the stochastic maximum principle for this kind of problem was first written by Djehiche et al [10], many works have been studied and continue this problem of risk-sensitive, we can mentioned as an example the papers of Chala [7,8,12].…”

Time-Optimal Control for Semilinear Stochastic Functional Differential Equations with Delays

“…Some standard optimal control results for general linear systems with delays in reflexive Banach spaces were studied in [13]. Recently, a survey of results on optimal control problems governed by delay differential inclusions was referred to by Mordukhovich et al [14,15], and [16,17]. Micu et al [18], who discussed the time-optimal boundary controls for the heat equation, and the parabolic equations with the Neumann condition was considered by Krakowiak [19].…”