On the numerical solution of some nonlinear stochastic differential equations using the semi-discrete method

TL;DR

This paper introduces a semi-discrete numerical scheme for solving super linear stochastic differential equations with non-negative solutions, ensuring positivity preservation and achieving an optimal strong convergence order of at least 1/2.

Contribution

It develops a new explicit numerical method that preserves positivity for super linear SDEs, including the Heston 3/2-model, improving upon existing schemes.

Findings

The semi-discrete method preserves positivity in super linear SDEs.

The method achieves at least 1/2 strong convergence order.

Numerical experiments confirm theoretical convergence results.

Abstract

In this paper we are interested in the numerical solution of stochastic differential equations with non negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super linear stochastic differential equations. It is well known that the usual Euler scheme diverges on super linear problems and the Tamed-Euler method does not preserve positivity. In that direction, we use the Semi-Discrete method that the first author has proposed in two previous papers. We propose a new numerical scheme for a class of stochastic differential equations which are super linear with non negative solution. In this class of stochastic differential equations belongs the Heston $3/2$-model that appears in financial mathematics, for which we prove %theoretically and through numerical experiments the "optimal" order of strong convergence at least $1/2$ of the Semi-Discrete method.