Mixed optimal control of forward-backward stochastic system

This paper is concerned with a mixed optimal control problem driven by forward-backward stochastic differential equation, where the term mixed means that there are two controllers, one is a deterministic controller and the other is a random controller. We derive a necessary condition and a sufficient condition for the mixed optimal control problem. A linear-quadratic mixed optimal control problem is discussed, and the corresponding optimal control is expressed by the solution of mean-field forward-backward stochastic differential equation. A feature here is that we obtain a mean-field forward-backward stochastic differential equation, which naturally arises from the study of mixed optimal control problem driven by forward-backward stochastic differential equation without mean-field term. As an application of theoretical results obtained here, we solve a problem of information security investment and cyber insurance.

Automated summary

This paper focuses on a mixed optimal control problem driven by forward-backward stochastic differential equation, discussing necessary and sufficient conditions, as well as solving a linear-quadratic mixed optimal control problem. The theoretical results are applied to solve a problem of information security investment and cyber insurance.