# A numerical approach to Kolmogorov equation in high dimension based on Gaussian analysis

## TL;DR

This paper introduces a novel numerical method for solving high-dimensional Kolmogorov equations by leveraging Gaussian analysis, providing an alternative to Monte Carlo methods with proven convergence.

## Contribution

The paper develops a Gaussian-based series expansion approach for high-dimensional Kolmogorov equations, inspired by SPDE structures, with convergence proof and numerical validation.

## Key findings

- Convergent series expansion for solutions in high dimensions.
- Numerical tests demonstrate efficiency over traditional methods.
- Applicable to infinite-dimensional stochastic systems.

## Abstract

For Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic Partial Differential Equations (SPDE) and thus contains an underlying Gaussian process which is the key of the algorithm. A series development of the solution in terms of iterated integrals of the Gaussian process is given, it is proved to converge - also in the infinite dimensional limit - and it is numerically tested in a number of examples.