Asymptotic error distribution for tamed Euler method with coupled monotonicity condition

TL;DR

This paper analyzes the asymptotic error distribution of a tamed Euler method for SDEs with coupled monotonicity, revealing how the regularization parameter affects long-term error behavior and extending results to additive noise cases.

Contribution

It establishes the asymptotic error distribution for the tamed Euler method with a coupled monotonicity condition and explores the impact of the regularization parameter on long-term errors.

Findings

The strong convergence rate is   rac{1}{2} for the proposed method.

The   .5 case yields the largest mean-square error over time.

Similar asymptotic error distributions are obtained for SDEs with additive noise.

Abstract

This paper establishes the asymptotic error distribution of the tamed Euler method for stochastic differential equations (SDEs) with a coupled monotonicity condition, that is, the limit distribution of the corresponding normalized error process. Specifically, for SDEs driven by multiplicative noise, we first propose a tamed Euler method parameterized by $\alpha\in (0, 1]$ and establish that its strong convergence rate is $\alpha\wedge\frac{1}{2}$. Notably, $\alpha $ can take arbitrary positive values by adjusting the regularization coefficient without altering the strong convergence rate. We then derive the asymptotic error distribution for this tamed Euler method. Further, we infer from the limit equation that among the tamed Euler method of strong order $\frac{1}{2}$, the one with $\alpha = \frac{1}{2}$ yields the largest mean-square error after a long time, while those of $\alpha>\frac{1}{2}$ share a unified asymptotic error distribution. In addition, our analysis is also extended to SDEs with additive noise and similar conclusions are obtained. Additional treatments are required to accommodate super-linearly growing coefficients, a feature that distinguishes our analysis on the asymptotic error distribution from established results.