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Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs

E. Gobet, J. G. López-Salas, P. Turkedjiev, C. Vázquez

TL;DR

This work introduces SRMDP, a stratified regression Monte Carlo framework to approximate the Y and Z components of discrete-time BSDEs and the associated semilinear PDEs, with a focus on enabling massive GPU parallelization and reducing memory demands. The key idea is stratified sampling within hypercubes of the state space and solving small, local regression problems for each stratum, which dramatically lowers memory usage while preserving convergence (the error scales like $O(N^{-1})$ when the number of local basis functions is fixed and the number of samples per stratum is appropriately chosen). The authors provide a rigorous non-asymptotic error analysis, optimality considerations for LP0/LP1 local bases, explicit OLS solutions suitable for GPU implementation, and a detailed complexity study showing substantial memory and time advantages over LSMDP, especially in high dimensions. Numerical experiments demonstrate significant GPU speedups and the practicality of solving high-dimensional BSDE/PDE problems, with detailed comparisons between LP0 and LP1 bases across dimensions up to $d=19$. The combination of stratified sampling, local regression, and GPU parallelism yields a scalable, memory-efficient approach for nonlinear PDEs and BSDEs, with broad implications for high-dimensional stochastic control and financial engineering problems.

Abstract

In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on many core processors such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization. In this way, we minimize the exposure to the memory requirements due to the storage of simulations. Indeed, we note the lower memory overhead of the method compared with previous works.

Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs

TL;DR

This work introduces SRMDP, a stratified regression Monte Carlo framework to approximate the Y and Z components of discrete-time BSDEs and the associated semilinear PDEs, with a focus on enabling massive GPU parallelization and reducing memory demands. The key idea is stratified sampling within hypercubes of the state space and solving small, local regression problems for each stratum, which dramatically lowers memory usage while preserving convergence (the error scales like when the number of local basis functions is fixed and the number of samples per stratum is appropriately chosen). The authors provide a rigorous non-asymptotic error analysis, optimality considerations for LP0/LP1 local bases, explicit OLS solutions suitable for GPU implementation, and a detailed complexity study showing substantial memory and time advantages over LSMDP, especially in high dimensions. Numerical experiments demonstrate significant GPU speedups and the practicality of solving high-dimensional BSDE/PDE problems, with detailed comparisons between LP0 and LP1 bases across dimensions up to . The combination of stratified sampling, local regression, and GPU parallelism yields a scalable, memory-efficient approach for nonlinear PDEs and BSDEs, with broad implications for high-dimensional stochastic control and financial engineering problems.

Abstract

In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on many core processors such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization. In this way, we minimize the exposure to the memory requirements due to the storage of simulations. Indeed, we note the lower memory overhead of the method compared with previous works.
Paper Structure (28 sections, 6 theorems, 61 equations, 13 tables, 4 algorithms)

This paper contains 28 sections, 6 theorems, 61 equations, 13 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. The problem
  3. Objectives
  4. New Regression Monte Carlo paradigm
  5. Literature review
  6. Notation
  7. Mathematical framework and basic properties
  8. Stratified algorithm and convergence results
  9. Algorithm
  10. Error analysis
  11. Proof of Theorem \ref{['thm:mc:err decomp']}
  12. GPU implementation
  13. Explicit solutions to OLS in Algorithm \ref{['alg:srmdp']}
  14. LP0
  15. LP1
  16. ...and 13 more sections

Key Result

Proposition 2.1

\newlabelprop:hnu Suppose that $\nu$ is the logistic distribution defined in $\bf (A_\nu)$. There is a constant $c_{\text{$\bf (A_\nu)$}}\in [1,+\infty)$ such that, for any function $h:\mathbb{R}^d\mapsto \mathbb{R}$ or $\mathbb{R}^q$ in $\mathbf{L}_2(\nu)$, for any $0\leq i\leq N$, and for any $i

Theorems & Definitions (13)

  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2: a.s. upper bounds, gobe:turk:14
  • Definition 3.1: Ordinary linear least-squares regression
  • Definition 3.2: Simulations and empirical measures
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Theorem 3.5: Error for the Stratified LSMDP scheme
  • Definition 3.6
  • ...and 3 more