Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients

TL;DR

This paper introduces a new class of explicit Euler schemes for stochastic differential equations with superlinearly growing coefficients, proving their convergence and providing rate estimates under mild conditions.

Contribution

It proposes explicit Euler schemes capable of handling superlinear growth in SDE coefficients, with proven convergence and convergence rate estimates.

Findings

Schemes converge in probability and in p to SDE solutions

Strong order  is achieved for p-convergence

Convergence holds under very mild conditions

Abstract

A new class of explicit Euler schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in probability and in $\mathcal{L}^p$ to the solution of the corresponding SDEs. Moreover, rate of convergence estimates are provided for $\mathcal{L}^p$ and almost sure convergence. In particular, the strong order $1/2$ is recovered in the case of uniform $\mathcal{L}^p$-convergence.