Boundary-preserving Lamperti-splitting schemes for some Stochastic Differential Equations

TL;DR

This paper introduces boundary-preserving numerical schemes for scalar SDEs with bounded state-space, utilizing Lamperti transform and Lie--Trotter splitting, achieving strong convergence and improved confinement.

Contribution

The paper develops and analyzes a novel Lamperti-splitting scheme that preserves boundaries and attains order 1 strong convergence for scalar SDEs with non-globally Lipschitz coefficients.

Findings

Schemes achieve $L^{p}(\Omega)$-convergence of order 1.

Numerical experiments confirm theoretical convergence rates.

Proposed schemes better confine solutions within the state-space compared to existing methods.

Abstract

We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove $L^{p}(\Omega)$-convergence of order $1$, for every $p \geq 1$, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.