Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems

SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page 1290-1303, December 2023.
Abstract. 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 [J. Han, A. Jentzen, and W. E, Proc. Natl. Acad. Sci. USA, 115 (2018), pp. 8505–8510] has shown great power in solving high-dimensional forward-backward stochastic differential equations and has inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which cannot be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [H. Wang et al., Deep Learning-Based BSDE Solver for LIBOR Market Model with Application to Bermudan Swaption Pricing and Hedging, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide a 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.