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

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.