随机最大值原理的时间反演
Time-Reversal of Stochastic Maximum Principle
AI总结：
针对FBSDE终端边界条件与正向信息流导致的数值求解困难，本文提出时间反演表示及迭代Monte-Carlo回归算法，并在LQ控制算例中验证。
摘要：
随机最大值原理（SMP）给出了随机最优控制问题解的必要条件。该条件涉及一个关于状态过程和伴随过程的正倒向随机微分方程（FBSDE）耦合系统。FBSDE的数值求解具有挑战性，因为伴随过程的边界条件是在终端时刻给定的，而解又必须适应于Wiener过程随时间正向的信息流。本文提出了FBSDE系统的一种“时间反演”，其中包含关于一个时间反向Wiener过程的积分。利用该时间反演，本文提出一种迭代Monte-Carlo程序，以同时求解FBSDE系统及其时间反演。该程序包括近似{Föllmer漂移}，并在每个时刻求解状态与其伴随量之间的回归问题。最后，通过一个数值算例在线性二次（LQ）最优控制问题上演示了该程序。
英文摘要：
Stochastic maximum principle (SMP) specifies a necessary condition for the solution of a stochastic optimal control problem. The condition involves a coupled system of forward and backward stochastic differential equations (FBSDE) for the state and the adjoint processes. Numerical solution of the FBSDE is challenging because the boundary condition of the adjoint process is specified at the terminal time, while the solution should be adaptable to the forward in time filtration of a Wiener process. In this paper, a "time-reversal" of the FBSDE system is proposed that involves integration with respect to a backward in time Wiener process. The time-reversal is used to propose an iterative Monte-Carlo procedure to solves the FBSDE system and its time-reversal simultaneously. The procedure involves approximating the {Föllmer's drift} and solving a regression problem between the state and its adjoint at each time. The procedure is illustrated for the linear quadratic (LQ) optimal control problem with a numerical example.
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