Quasi-Regression Monte-Carlo scheme for semi-linear PDEs and BSDEs with large scale parallelization on GPUs
E. Gobet, J. G. López-Salas, C. Vázquez
TL;DR
The paper develops a quasi-regression Monte Carlo scheme to numerically solve decoupled forward–backward SDEs and the related semi-linear PDEs by projecting onto a smooth, orthonormal basis and applying a weighted dynamic programming update. It leverages a sampling distribution $ u$ (e.g., a Student’s $t$-based measure) to obtain stability and Fourier-type error control, yielding non-asymptotic error bounds. The method is designed for massive GPU parallelization, with error propagation analyzed and supported by numerical experiments across moderate to high dimensions. This work addresses the curse of dimensionality in BSDE/PDE problems by combining a smooth quasi-regression framework with scalable hardware acceleration, enabling accurate high-dimensional solutions in practice.
Abstract
In this article we design a novel quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic differential equations (BSDEs), and we analyze the convergence of the proposed method. The algorithm also approximates the solution to the related semi-linear parabolic partial differential equation (PDE) obtained through the well-known Feynman-Kac representation. For the sake of enriching the algorithm with high-order convergence a weighted approximation of the solution is computed and appropriate conditions on the parameters of the method are inferred. With the challenge of tackling problems in high dimensions we propose suitable projections of the solution and efficient parallelizations of the algorithm taking advantage of powerful many-core processors such as graphics processing units (GPUs).