Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise

Xiang Li, Yingjun Mo, Haoran Yang

TL;DR

The paper addresses long-time numerical approximation of SDEs with non-globally Lipschitz drift and multiplicative noise using a Newton-based tamed Euler–Maruyama scheme with variable step sizes. By establishing moment bounds, one-step error estimates, and gradient bounds for the SDE semigroups, and applying a domino decomposition together with Malliavin-type techniques, the authors prove uniform-in-time convergence in $\mathbb{W}_1$ and $d_{TV}$ at a rate $C\eta_n^{\alpha}$ for any $\alpha\in(0,\tfrac12)$. The results rely on partial dissipativity (ergodicity) assumptions and careful control of the taming, enabling explicit rates and applicability to additive noise cases as well. The work provides a practically implementable long-time scheme with provable convergence guarantees for challenging SDEs with super-linear drift, supported by detailed moment, one-step, and gradient analyses. Overall, the paper contributes a rigorous framework for stable long-time simulation of non-globally Lipschitz SDEs under multiplicative noise, with explicit rate information dependent on the step-size sequence $\{\eta_n\}$.

Abstract

Consider the following stochastic differential equation driven by multiplicative noise on $\mathbb{R}^d$ with a superlinearly growing drift coefficient, \begin{align*} \mathrm{d} X_t = b (X_t) \, \mathrm{d} t + σ(X_t) \, \mathrm{d} B_t. \end{align*} It is known that the corresponding explicit Euler schemes may not converge. In this article, we analyze an explicit and easily implementable numerical method for approximating such a stochastic differential equation, i.e. its tamed Euler-Maruyama approximation. Under partial dissipation conditions ensuring the ergodicity, we obtain the uniform-in-time convergence rates of the tamed Euler-Maruyama process under $L^{1}$-Wasserstein distance and total variation distance.

Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise

TL;DR

The paper addresses long-time numerical approximation of SDEs with non-globally Lipschitz drift and multiplicative noise using a Newton-based tamed Euler–Maruyama scheme with variable step sizes. By establishing moment bounds, one-step error estimates, and gradient bounds for the SDE semigroups, and applying a domino decomposition together with Malliavin-type techniques, the authors prove uniform-in-time convergence in and at a rate for any . The results rely on partial dissipativity (ergodicity) assumptions and careful control of the taming, enabling explicit rates and applicability to additive noise cases as well. The work provides a practically implementable long-time scheme with provable convergence guarantees for challenging SDEs with super-linear drift, supported by detailed moment, one-step, and gradient analyses. Overall, the paper contributes a rigorous framework for stable long-time simulation of non-globally Lipschitz SDEs under multiplicative noise, with explicit rate information dependent on the step-size sequence .

Abstract

Consider the following stochastic differential equation driven by multiplicative noise on with a superlinearly growing drift coefficient, \begin{align*} \mathrm{d} X_t = b (X_t) \, \mathrm{d} t + σ(X_t) \, \mathrm{d} B_t. \end{align*} It is known that the corresponding explicit Euler schemes may not converge. In this article, we analyze an explicit and easily implementable numerical method for approximating such a stochastic differential equation, i.e. its tamed Euler-Maruyama approximation. Under partial dissipation conditions ensuring the ergodicity, we obtain the uniform-in-time convergence rates of the tamed Euler-Maruyama process under -Wasserstein distance and total variation distance.
Paper Structure (11 sections, 12 theorems, 171 equations)

This paper contains 11 sections, 12 theorems, 171 equations.

Table of Contents

  1. Introduction and Main Results
  2. Notations
  3. Assumptions and main Results
  4. Auxiliary Lemmas
  5. Moment estimates
  6. One step error estimates
  7. Gradient estimate for the semigroups of $X_t$
  8. Proof of Main Results
  9. The estimate of the last step
  10. Proof of main results
  11. Technical lemmas

Key Result

Theorem 1.1

Let $\left(X_t\right)_{t \geqslant 0}$ and $\left({Y}_k\right)_{k \geqslant 0}$ be defined by SDE and EM1. Suppose Assumption A1, A2, and A3 hold with $\eta_1 \leqslant \eta$ and $\theta \leqslant \theta_0$, and $b \in \mathcal{C}^2 (\mathbb{R}^d; \mathbb{R}^d)$ satisfies Then for any $\alpha \in (0, 1 / 2)$, there exists a constant $C > 0$ such that, where $C$, $\eta>0$, and $\theta_0>0$ only

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2: Additive case
  • Lemma 2.1: Moment estimates for $X_t$
  • proof
  • Lemma 2.2
  • Lemma 2.3: Moment estimates for $Y_{t_n}$
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5: Bismut–Elworthy–Li formula
  • ...and 13 more