A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations
Lorenc Kapllani, Long Teng
TL;DR
This work introduces a forward differential deep learning method to solve high-dimensional nonlinear BSDEs by recasting the problem through Malliavin calculus into a differential learning task for the triple $(Y, Z, \Gamma)$. Discretization via Euler-Maruyama coupled with three neural networks enables global optimization with a differential loss that directly encodes the discretized dynamics. The proposed DLDBSDE scheme generalizes prior forward approaches by incorporating Malliavin derivatives and a weighted loss that improves first- and second-order gradient accuracy, notably reducing computation time. Across diverse high-dimensional tests, including 1D and 50D cases, the method achieves higher accuracy for $Y$, $Z$, and $\Gamma$ than existing forward schemes, with strong implications for pricing and hedging in finance. The results demonstrate the practical potential of DLDBSDE as an efficient tool for solving high-dimensional stochastic control problems.
Abstract
In this work, we present a novel forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). Motivated by the fact that differential deep learning can efficiently approximate the labels and their derivatives with respect to inputs, we transform the BSDE problem into a differential deep learning problem. This is done by leveraging Malliavin calculus, resulting in a system of BSDEs. The unknown solution of the BSDE system is a triple of processes $(Y, Z, Γ)$, representing the solution, its gradient, and the Hessian matrix. The main idea of our algorithm is to discretize the integrals using the Euler-Maruyama method and approximate the unknown discrete solution triple using three deep neural networks. The parameters of these networks are then optimized by globally minimizing a differential learning loss function, which is novelty defined as a weighted sum of the dynamics of the discretized system of BSDEs. Through various high-dimensional examples, we demonstrate that our proposed scheme is more efficient in terms of accuracy and computation time compared to other contemporary forward deep learning-based methodologies.