Path-Dependent Deep Galerkin Method: A Neural Network Approach to Solve Path-Dependent Partial Differential Equations

In this paper, we propose a novel numerical method for path-dependent partial differential equations (PPDEs). These equations first appeared in the seminal work of [B. Dupire, Quant. Finance, 2019 (2009), pp. 721--729], where the functional Ito <^>\ calculus was developed to deal with path-dependent financial derivatives. More specifically, we generalize the deep Galerkin method (DGM) of [J. Sirignano and K. Spiliopoulos, J. Comput. Phys., 375 (2018), pp. 1339--1364] to deal with these equations. The method, which we call path-dependent DGM, consists of using a combination of feed-forward and long short-term memory architectures to model the solution of the PPDE. We then analyze several numerical examples, many from the financial mathematics literature, that show the capabilities of the method under very different situations.