Asymptotic error distribution for tamed Euler method with coupled monotonicity condition

TL;DR

This work analyzes the asymptotic error distribution of the tamed Euler method for SDEs under a coupled monotone condition that allows super-linear growth. It introduces a parameterized scheme with regularization exponent $α$, proving a strong convergence rate of $α\wedge\frac{1}{2}$ for multiplicative noise and $α\wedge1$ for additive noise, with $α$ freely chosen via regularization. The core contribution is establishing the asymptotic distribution of the normalized error: for multiplicative noise, $N^{\frac{1}{2}\wedge α}(\bar{X}_N^{α}-X(T))$ converges to a linear-SDE-based limit $U^α$, whose form depends on whether $α\le\tfrac{1}{2}$ or $α\ge\tfrac{1}{2}$; for additive noise, a parallel limit $V^α$ is obtained with $α$-dependent drift and diffusion corrections, including higher-order terms for $α\ge1$. The results are supported by rigorous probabilistic techniques (stable convergence, Jacod-type limit theorems) and validated by numerical experiments, which illustrate the α-dependent error structures and confirm the predicted convergence rates. Overall, the paper extends the theory of asymptotic error distributions to tamed Euler schemes under coupled monotone growth and provides guidance on parameter choices to control long-time error behavior in SDE simulations.

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 $α\in (0, 1]$ and establish that its strong convergence rate is $α\wedge\frac{1}{2}$. Notably, $α$ 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 $α= \frac{1}{2}$ yields the largest mean-square error after a long time, while those of $α>\frac{1}{2}$ 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.

Figures (4)

• Figure 1: Mean square errors for tamed Euler methods \ref{['TEM']} with $\alpha = 0.2$, $0.4$, $0.5$, and $1$ applied to \ref{['SDEE1']} in the log-log scale for five different step-sizes $h = 2^{-8}, 2^{-9}, 2^{-10}, 2^{-11}, 2^{-12}$.
• Figure 2: Evolution of mean-square errors for tamed Euler methods \ref{['TEM']} with $\alpha = 0.5$, $0.7$, $0.8$, and $1$ applied to \ref{['SDEE1']} with $h = 10^{-2}$.
• Figure 3: Mean square errors for tamed Euler methods \ref{['TEMA']} with $\alpha = 0.2$, $0.7$, $1$, $1.5$, and $2$ applied to \ref{['SDEE2']} in the log-log scale for five different step-sizes $h = 2^{-6}, 2^{-7}, 2^{-8}, 2^{-9}, 2^{-10}$.
• Figure 4: Evolution of mean-square errors for tamed Euler methods \ref{['TEMA']} with $\alpha = 1$, $1.5$, $2$, and $2.5$ applied to \ref{['SDEE2']} with $h = 10^{-2}$.

Theorems & Definitions (43)

• Proposition 2.1
• Proposition 2.2
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Corollary 3.5