Logarithmic Euler Maruyama Scheme for Multi Dimensional Stochastic Delay Differential Equation

TL;DR

This paper extends the logarithmic Euler-Maruyama scheme to multi-dimensional stochastic delay differential equations, ensuring positivity and demonstrating a convergence rate of 0.5, which improves numerical solutions for such systems.

Contribution

It introduces a new scheme for multi-dimensional stochastic delay differential equations that guarantees positivity and proves its convergence rate.

Findings

Scheme maintains positivity under initial conditions

Convergence rate of 0.5 established

Applicable to multi-dimensional systems

Abstract

In this paper, we extend the logarithmic Euler-Maruyama scheme for stochastic delay differential equation in one dimension to the part where we propose a scheme for a system of stochastic delay differential equations. We then show that the scheme always maintains positivity subject to initial conditions. We then show the convergence of the proposed Euler-Maruyama scheme. With this scheme, all the approximate solutions are positive and the rate of convergence of this scheme is 0.5.