A Comparison of Numerical Methods for Solving Backward Stochastic Differential Equations

Abstract

We study several different numerical methods for solving Backwards Stochastic Differential Equations and Partial Differential Equations. The main methods reviewed are a least-squares Monte-Carlo method, and a method utilizing Artificial Neural Networks: the Deep BSDE method. We implement both algorithms and compare their performance solving a BSDE to find the fair price of a financial option in an incomplete market. We find that the Deep BSDE method can provide similar approximation efficiency to the Monte-Carlo algorithm with a limited number of partition points, but outperforms it as the number of partition points increases.

In addition, we examine another technique that utilizes artificial neural networks to solve Partial Differential Equations directly, the Deep PDE Method. This method provides solution dynamics across the entire domain of the problem, as opposed to the Deep BSDE method which provides solution dynamics only for a small subset of points in the domain. This method is computationally intensive to train, and also requires bounded domains and boundary conditions and bounded domains, unlike the Deep BSDE algorithm where the domain is unbounded. It remains a future research question to see whether the Deep BSDE method could be modified to work with bounded domains or if the Deep PDE Method could somehow be expanded to work on unbounded domains.