A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations

Nevertheless, a large class of PPDE is not analytically solvable, and one has to rely on the numerical solution. In Ren and Tan (2017), a probabilistic scheme based on Fahim et al. (2011) has been proposed, and was proved to converge to the viscosity solution of PPDE. However, its practical implementation is far from trivial due to the presence of the conditional expectation. The suggestion of Ren and Tan (2017) to use regression as in Gobet et al. (2005) relies on a careful basis function choice, which may not always be possible, see the discussion at the end of Section 2. Neural networks methods for PDEs have been introduced independently in Han et al. (2018) and Sirignano and Spiliopoulos (2018) using backward stochastic differential equations and the Galerkin method respectively, see also Beck et al. (2019), Huré et al. (2020) for other variants of deep learning-based numerical solutions. A deep neural network algorithm for the numerical solution of PPDEs has also been proposed in Saporito and Zhang (2020) by applying Long Short-Term Memory (LSTM) networks in the framework of the deep Galerkin method for PDEs, see Sirignano and Spiliopoulos (2018). On the other hand, Sabate-Vidales et al. (2020) propose to combine the LSTM net-2 work and the path signature to solve the linear PPDE. Unlike regression methods, deep learning algorithms do not rely on the choice of a basis. In this paper, we propose a deep learning approach to the implementation of the probabilistic scheme of Ren and Tan (2017) for the numerical solution of fully nonlinear PPDEs of the form (1.1). The main idea of Algorithm 4.1 is based on the L