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.

中文翻译：

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前后随机系统的混合最优控制

本文关注的是由前向-后向随机微分方程驱动的混合最优控制问题，其中“混合”一词意味着有两个控制器，一个是确定性控制器，另一个是随机控制器。我们推导了混合最优控制问题的必要条件和充分条件。讨论了线性二次混合最优控制问题，并用均值前后正向随机微分方程的解表示了相应的最优控制。这里的一个特点是，我们获得了均值正向-后向随机微分方程，这自然是由于研究由无均值场项的正向-后向随机微分方程驱动的混合最优控制问题而产生的。作为此处获得的理论结果的应用，