Abstract
This paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations with Lévy process (FBSDEL). We derive a necessary condition for the existence of the optimal control by means of spike variational technique, while the control domain is not necessarily convex. Simultaneously, we also get the maximum principle for this control system when there are some initial and terminal state constraints. Finally, a financial example is discussed to illustrate the application of our result.
Highlights
Stochastic optimal control is an important matter that cannot be neglected in modern control theory in long days
Shi and Wu [8] and Shi [9] acquired the maximum principle for a kind of forward-backward stochastic control system with Poisson jumps in the form of local and global, respectively. e fully coupled forward-backward stochastic control system was extended by Liu et al [10] at the base of Shi and Wu [8], and in the they obtained the maximum principle with the control system be constrained about initial-terminal state constraints
We will study the optimal control problem for forward-backward stochastic control systems driven by Levy process, which could be considered as a nonconvex control domain case that is extended from the result of [25]
Summary
Stochastic optimal control is an important matter that cannot be neglected in modern control theory in long days. Wang and Wu [14] proposed a backward separation approach and replaced the original state and observation equation with the Zakai equation, Mathematical Problems in Engineering and lots of complicated stochastic calculi in infinite-dimensional spaces were avoided in this way Based on this approach, Xiao [15] studied a partially observed optimal control of forwardbackward stochastic systems with random jumps and obtained the maximum principle and sufficient conditions of an optimal control under some certain convexity assumptions. We will study the optimal control problem for forward-backward stochastic control systems driven by Levy process, which could be considered as a nonconvex control domain case that is extended from the result of [25].
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