Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods, Split-Step and Tamed Schemes

TL;DR

This work addresses time-uniform weak convergence of numerical schemes for SDEs with non-globally Lipschitz drifts by developing a general UiT criterion grounded in Strong Exponential Stability (SES) of the SDE semigroup and uniform moment control of discretisations. It then applies this framework to several schemes: Euler–Maruyama under global Lipschitz assumptions (β = 1/2), split-step and implicit Euler via a modified SDE with δ-independent SES, and a truncated tamed Euler scheme, each yielding UiT bounds that hold for all times. The paper provides explicit, verifiable conditions on the drift and diffusion that guarantee UiT convergence and illustrates these results with examples and discussions on stability relative to initial data. Overall, the results offer robust, non-asymptotic guarantees for long-time simulations of SDEs in settings with locally Lipschitz drift, without requiring knowledge of invariant measures, thereby enhancing reliability for transients and equilibria alike.

Abstract

We prove a general criterion providing sufficient conditions under which a time-discretiziation of a given Stochastic Differential Equation (SDE) is a uniform in time approximation of the SDE. The criterion is also, to a certain extent, discussed in the paper, necessary. Using such a criterion we then analyse the convergence properties of numerical methods for solutions of SDEs; we consider Explicit and Implicit Euler, split-step and (truncated) tamed Euler methods. In particular, we show that, under mild conditions on the coefficients of the SDE (locally Lipschitz and strictly monotonic), these methods produce approximations of the law of the solution of the SDE that converge uniformly in time. The theoretical results are verified by numerical examples.

Figures (1)

• Figure 1: Illustration of standard tamed Euler \ref{['standard_tamed']} and truncated tamed Euler schemes \ref{['tamed']} (for different constants, namely $\alpha=1, 1.3, 5$) with step $\delta=0.05$. We plot time-approximations of $\mathbb{E}[x_t]$, with $x_t$ solution of $dx_t = -(x^3+x)dt+dB_t$, with initial state $x_0=1$ (left) and $x_0=100$ (right), taking averages over 1000 samples. The solution is not explicitly computable so the benchmark 'theoretical line' is obtained using the standard tamed Euler scheme \ref{['standard_tamed']} with $\delta=5\times 10^{-4}$ and using $10^4$ samples. For small initial data the choice of $\alpha$ in TTE does not matter and all the schemes behave substantially in the same way; for large initial data $\alpha$ in TTE needs to be chosen compatibly with \ref{['tamed_edelta']}, see Note \ref{['note:section5']}. The choice $\alpha=1$ violates \ref{['tamed_edelta']}, giving rise to the incorrect non-smooth behaviour of the right panel (blue line). Within the range of $\alpha$ satisfying \ref{['tamed_edelta']}, TTE seems to always outperform the standard tamed scheme.

Theorems & Definitions (44)

• Proposition 2.2
• Lemma 2.4
• proof : Proof of Proposition \ref{['prop_weak']}
• Example 2.5
• Remark 2.7
• Lemma 2.8
• Remark
• Remark
• Lemma 2.10
• Proposition 3.2