Strong Convergence of a GBM Based Tamed Integrator for SDEs and an Adaptive Implementation

TL;DR

This paper presents a new tamed exponential integrator for SDEs with one-sided Lipschitz drift, proving strong convergence and demonstrating its efficiency and adaptability in various models.

Contribution

The paper introduces a novel GBM-based tamed integrator with proven strong convergence and extends it to nonlinear diffusion, enhancing numerical methods for SDEs.

Findings

Order 1 convergence for linear diffusion terms

Efficient compared to fixed step methods

Effective in adaptive time stepping schemes

Abstract

We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches are illustrated by considering some well-known SDE models.