Positivity preserving logarithmic Euler-Maruyama type scheme for stochastic differential equations

TL;DR

This paper introduces explicit numerical schemes that preserve positivity for stochastic differential equations, providing convergence analysis and numerical validation for equations with exponential growth coefficients.

Contribution

It develops a new class of positivity-preserving Euler-Maruyama schemes with proven convergence and rate results for challenging SDEs.

Findings

Numerical schemes maintain positivity of solutions.

Convergence and rate results are established.

Numerical experiments confirm theoretical findings.

Abstract

In this paper, we propose a class of explicit positivity preserving numerical methods for general stochastic differential equations which have positive solutions. Namely, all the numerical solutions are positive. Under some reasonable conditions, we obtain the convergence and the convergence rate results for these methods. The main difficulty is to obtain the strong convergence and the convergence rate for stochastic differential equations whose coefficients are of exponential growth. Some numerical experiments are provided to illustrate the theoretical results for our schemes.