Some stochastic process techniques applied to deterministic models

TL;DR

This paper explores stochastic process techniques to analyze exit times of stochastic systems within bounded domains, extending solutions to higher dimensions and demonstrating the effectiveness of a numerical scheme in FreeFEM.

Contribution

It introduces a numerical scheme for computing exit times in multidimensional stochastic systems, extending existing one-dimensional solutions and bridging theory with computational methods.

Findings

Effective numerical scheme implemented in FreeFEM

Extension of one-dimensional solutions to higher dimensions

Illustrative examples demonstrating theoretical results

Abstract

Stochastic mathematical models are essential tools for understanding and predicting complex phenomena. The purpose of this work is to study the exit times of a stochastic dynamical system-specifically, the mean exit time and the distribution of exit times of the stochastic process within a bounded domain. These quantities are obtained by solving elliptic and parabolic partial differential equations (PDEs), respectively. To support practical applications, we propose a numerical scheme implemented in FreeFEM, emphasizing its effectiveness in two- and three-dimensional cases due to the software's limitations in higher dimensions. The examples provided illustrate the theoretical results, which extend known one-dimensional solutions to higher-dimensional settings. This contribution bridges theoretical and computational approaches for analyzing stochastic processes in multidimensional domains, offering insights into their behavior and potential applications.