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

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. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper proposes a class of explicit positivity preserving numerical methods for general stochastic differential equations with positive solutions. The convergence and convergence rate results for these methods are obtained under certain reasonable conditions. The main challenge lies in obtaining strong convergence and convergence rate for stochastic differential equations with coefficients of exponential growth. Numerical experiments are provided to illustrate the theoretical results for the proposed schemes.