Numerical computation of probabilities for nonlinear SDEs in high dimension using Kolmogorov equation

TL;DR

This paper develops a numerical method for high-dimensional nonlinear SDEs by solving the associated Kolmogorov equations, improving accuracy through a Gaussian process shift, and achieves good results in dimensions around 100.

Contribution

It introduces an enhanced numerical approach combining Kolmogorov equation solutions with a Gaussian shift, improving results for high-dimensional nonlinear SDEs.

Findings

Effective in dimensions around 100

Significant improvement over previous methods

Combines Monte Carlo with Kolmogorov equation solutions

Abstract

Stochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to compute numerically expected values and probabilities associated to their solutions, by solving the associated Kolmogorov equations, with a partial use of Monte Carlo strategy - precisely, using Monte Carlo only for the linear part of the SDE. The basic idea was presented in Flandoli et al., JMAA (2020), but here we strongly improve the numerical results by means of a shift of the auxiliary Gaussian process. For relatively simple nonlinearities, we have good results in dimension of the order of 100.