论文标题
适用于随机控制的强大深FBSDE方法的收敛性
Convergence of a robust deep FBSDE method for stochastic control
论文作者
论文摘要
在本文中，我们提出了一种基于深度学习的数值方案，用于强烈耦合FBSDE，这是由随机控制引起的。这是对深BSDE方法的修改，其中向后方程的初始值不是一个免费参数，并且新的损失函数是控制问题成本的加权总和，而差异项与终端条件下的平均平方误差相吻合。我们通过一个数字示例表明，经典深度BSDE方法的直接扩展为FBSDE，失败了简单的线性季度控制问题，并激励新方法为何工作。在定期和有限性的假设上，对时间连续和时间离散控制问题的确切控制，我们为我们的方法提供了错误分析。我们从经验上表明，该方法会收敛三个不同的问题，一个方法是直接扩展深BSDE方法的问题。
In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.
