论文标题
向后深BSDE方法与应用程序的融合到最佳停止问题
Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems
论文作者
论文摘要
最佳的停止问题是金融市场中的核心问题之一，其价格广泛，例如美国和百慕大的选择。 Deep BSDE方法[Han，Jentzen和E，PNAS，115（34）：8505-8510，2018]在求解高维向前偏移的随机微分方程（FBSDES）方面表现出很大的力量，并激发了许多应用。但是，该方法以前向方式求解向后的随机微分方程（BSDE），该方程式无法用于最佳停止问题，而通常需要向后运行BSDE。为了克服这一困难，最近的一篇论文[Wang，Chen，Sudjianto，Liu和Shen，Arxiv：1807.06622，2018]提出了向后深的BSDE方法来解决最佳的停止问题。在本文中，我们为向后深的BSDE方法提供了严格的理论。具体而言，1。我们得出A后验误差估计，即，数值解的误差可以由训练损耗函数界定；和; 2。我们给出了损耗函数的上限，这可能足够小，可以通过普遍近似。我们给出了两个数值示例，这些示例与证明的理论相呈一致的表现。
The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward-backward stochastic differential equations (FBSDEs), and inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide the rigorous theory for the backward deep BSDE method. Specifically, 1. We derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and; 2. We give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory.
