A probabilistic interpretation of the parametrix method

TL;DR

This paper presents a probabilistic interpretation of the parametrix method for constructing fundamental solutions, enabling Monte Carlo simulations for diffusions and jump processes with H"{o}lder continuous coefficients.

Contribution

It introduces a novel probabilistic perspective on the parametrix technique, expanding its applicability to stochastic differential equations with irregular coefficients.

Findings

Provides a probabilistic framework for the parametrix method.

Demonstrates Monte Carlo simulation applicability.

Applies to diffusions and jump-driven SDEs with H"{o}lder continuous coefficients.

Abstract

In this article, we introduce the parametrix technique in order to construct fundamental solutions as a general method based on semigroups and their generators. This leads to a probabilistic interpretation of the parametrix method that is amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with H\"{o}lder continuous coefficients.