Quasi-Regression Monte-Carlo scheme for semi-linear PDEs and BSDEs with large scale parallelization on GPUs

TL;DR

This paper introduces a novel quasi-regression Monte Carlo algorithm for solving high-dimensional semi-linear PDEs and BSDEs, leveraging GPU parallelization for efficiency and high-order convergence.

Contribution

The paper presents a new quasi-regression Monte Carlo method with high-order convergence and GPU parallelization for high-dimensional semi-linear PDEs and BSDEs.

Findings

Convergence analysis of the proposed algorithm.

Effective high-dimensional problem handling with GPU acceleration.

High-order approximation achieved through weighted solutions.

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).