论文标题
前向后随机微分方程的最佳控制：时间含义和时间一致的解决方案
Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions
论文作者
论文摘要
本文涉及前向后的随机微分方程（简称FBSDE）的最佳控制问题，其递归成本函数由向后的随机Volterra积分方程（简称BSVIE）确定。发现这样一个最佳控制问题通常是时间不一致的，即使成本功能降低到经典的Bolza型，如Peng [50]，Lim-Zhou [41]和Yong [74]。因此，我们将寻找一种时间一致且局部最佳的平衡策略，而不是找到全球最佳控制（这是时间的一致性），可以通过相关的汉密尔顿 - 贾科比·贝尔曼（HJB）方程来构建该策略。验证定理的平衡策略局部最优性通过广义Feynman-kac公式用于BSVIES，并证明了抛物线抛物线部分微分方程（PDES，简称PDE）的某些稳定性估计。在某些条件下，证明是非局部PDE的平衡HJB方程，它可以接受独特的经典解决方案。作为特殊案例和应用，线性季节问题，均值变化模型，异质性爱泼斯坦 - Zin实用程序的社会计划者问题以及Stackelberg游戏。事实证明，我们的框架不仅可以涵盖[50,41,74]中研究的FBSDE的最佳控制问题，而且还可以涵盖在Yong [75,77]和Björk-Khapko-Murgoci [75,77]中研究的一般折现和对终末状态有条件期望的某些非线性外观的问题。
This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [50], Lim-Zhou [41], and Yong [74]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman-Kac formula for BSVIEs and some stability estimates of the representation for parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein-Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [50,41,74], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [75,77] and Björk-Khapko-Murgoci [7].
