Drift-implicit Euler scheme for sandwiched processes driven by H\"older noises

TL;DR

This paper studies the convergence of a drift-implicit Euler scheme for stochastic differential equations driven by H"older continuous noises, demonstrating a convergence rate equal to the H"older exponent under mild conditions.

Contribution

It establishes the convergence rate of the drift-implicit Euler scheme for SDEs driven by arbitrary H"older noises, extending previous results to unbounded drifts and general models.

Findings

Convergence rate equals the H"older exponent under mild assumptions.

Numerical schemes for generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models are analyzed.

Simulations confirm theoretical convergence rates.

Abstract

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary $\lambda$-H\"older continuous process, $\lambda\in(0,1)$. We prove that, under some mild moment assumptions on the H\"older constant of the noise, the $L^r(\Omega;L^\infty([0,T]))$-rate of convergence is equal to $\lambda$. To exemplify, we consider numerical schemes for the generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models. The results are illustrated by simulations.