Analysis of a first-order explicit positivity preserving scheme for a class of scalar SDEs

Abstract

We propose and analyze a first-order numerical scheme for a class of scalar Itô stochastic differential equations with almost surely positive solutions. We construct a new explicit numerical scheme, such that for any choice of the time-step size, the numerical solution is guaranteed to remain almost surely positive. The main result of this article is the first-order strong convergence of the proposed positivity preserving scheme, which is illustrated with numerical experiments and proved rigorously. It is also shown how to adapt the scheme and the results to Stratonovich stochastic differential equations with positive solutions.