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

Haoran Yang; Xiang Li; Yingjun Mo

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

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Pith's one-line read This paper proves that a tamed Euler–Maruyama scheme for SDEs with non-globally Lipschitz drift and multiplicative noise has uniform-in-time convergence rates of order $\eta_n^\alpha$ under the $L^1$-Wasserstein and total-variation…

desk verdict Genuinely new infinite-horizon W1/TV rates for tamed EM with multiplicative noise; proof holds up, main caveat is the strong but explicit uniform ellipticity assumption on sigma. read the letter →

arxiv 2505.03104 v1 pith:N7HL6T6O submitted 2025-05-06 math.PR

Xiang Li , Yingjun Mo , Haoran Yang This is my paper · ORCID

classification math.PR MSC 60H1065C3060J60

keywords SDEswithpolynomiallygrowingdrifttamedEuler-MaruyamaschemedecreasingstepWassersteindistancetotalvariationconvergenceratemultiplicativenoiseuniform-in-time

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that a simple, explicit tamed Euler–Maruyama scheme—where the drift increment is divided by $1+\eta_n^\alpha\|\nabla b\|_{\mathrm{op}}$—approximates the true process of an SDE with a superlinearly growing, dissipative drift and multiplicative noise not only on finite intervals but uniformly over the whole infinite time horizon. The main theorems give, for any $\alpha\in(0,1/2)$, a constant $C$ such that both the $L^1$-Wasserstein distance and the total-variation distance between the law of the true solution at time $t_n$ and the law of the scheme are at most $C\eta_n^\alpha$ for every $n$. The rates are obtained for variable step sizes satisfying a mild decay condition, and the additive-noise case is covered as a separate theorem. The significance is that the tamed scheme is explicit and easy to implement, so it offers a practical way to simulate the long-time distributional behavior of SDEs whose drift lies outside the globally Lipschitz class.

What carries the argument

The proof is carried by the domino decomposition, which telescopes the total discrepancy into a sum over steps: $P_{0,t_n}f(x_0)-Q_{0,t_n}f(x_0)=\sum_{k=1}^n Q_{0,t_{k-1}}(P_{t_{k-1},t_k}-Q_{t_{k-1},t_k})P_{t_k,t_n}f(x_0)$. The paper combines this decomposition with gradient estimates for the continuous semigroup obtained from the Bismut–Elworthy–Li integration-by-parts identity, where the uniform ellipticity of $\sigma$ gives a bound on $\sigma^{-1}$ used in that identity, and with a Gaussian one-step error estimate for the tamed step. The gradient estimates decay exponentially in $t_n-t_k$, so the sum over $k$ is controlled by a new summation lemma (Lemma 3.2) that converts the time-decaying terms into an overall $\eta_n^\alpha$. Moment estimates for both processes at all orders, via the exponential Lyapunov function $V(x)=e^{|x|}$ near infinity, keep the unbounded polynomial factors under control at every time.

What would settle it

Check whether the claimed uniform-in-time rate persists when the assumption on $\sigma$ is minimally violated: take a one-dimensional SDE with a dissipative polynomial drift, e.g. $b(x)=-x^3-x$, and a bounded but not uniformly elliptic diffusion such as $\sigma(x)=e^{-x^2}$, run the tamed scheme with $\eta_n=n^{-1/2}$, and measure $\sup_n W_1(\mathcal L(X_{t_n}),\mathcal L(Y_{t_n}))//\eta_n^\alpha$ for a fixed $\alpha\in(0,1/2)$; if the ratio diverges with $n$, the uniform-in-time claim genuinely depends on uniform ellipticity, while if it stays bounded, the assumption is stronger than necessary.

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Extended reading notes

Core claim

The paper's central claim is that the taming denominator $1+\eta_n^\alpha\|\nabla b(Y_{t_{n-1}})\|_{\mathrm{op}}$ in the Euler increment is enough to control the superlinear drift for all times, not just up to a fixed horizon. Under the three assumptions (dissipative polynomial drift, uniformly bounded and uniformly elliptic diffusion with bounded first and second derivatives, and variable step sizes that are nonincreasing, tend to zero, have divergent sum, and satisfy $\eta_{n-1}-\eta_n\le \theta\eta_n^2$), and with a $C^2$ drift whose Hessian grows at most polynomially, the discrepancy between the true Markov semigroup and the discrete semigroup is bounded by $C\eta_n^\alpha$ in both $W_1$ and total variation, uniformly in $n$. The additive-noise version relaxes the one-step error but keeps the same conclusion.

Load-bearing premise

The proof needs the diffusion matrix $\sigma$ to be globally bounded, uniformly elliptic ($\sigma^{-1}$ uniformly bounded), and twice continuously differentiable; if the multiplicative noise can vanish, grow, or have unbounded derivatives, the gradient estimates and the total-variation control collapse.

Editorial extensions

If this is right

The tamed scheme is uniformly accurate in distribution over the infinite horizon: at any fixed large $n$, the law of the discrete process is within $C\eta_n^\alpha$ of the law of the SDE in both Wasserstein and total-variation distance, with a constant independent of $n$.
Variable step sizes are allowed: any nonincreasing sequence $\eta_n\downarrow 0$ with divergent sum and step-difference bound $\eta_{n-1}-\eta_n\le\theta\eta_n^2$, such as $\eta_n=\eta/n^\gamma$, inherits the same rate.
For additive noise $\sigma\equiv\sigma_0$, the same result holds, and the proof uses the sharper one-step error $O(\eta_k^{2+2\alpha})$ for intermediate steps.
The uniform bound applies at every discrete time $t_n$, so long-time simulations with the tamed scheme remain distributionally close to the true solution at every step rather than drifting away as $n$ grows.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Combining the paper's bounds with the exponential ergodicity of the true SDE (which its lemmas establish) implies that the law of $Y_{t_n}$ reaches a neighborhood of the invariant measure at the same rate $\eta_n^\alpha$; the paper does not frame the result this way, but it makes the tamed scheme a viable explicit sampler for the ergodic limit.
If the uniform ellipticity of $\sigma$ were relaxed to a hypoelliptic or degenerate setting, the total-variation conclusion would likely fail because the Gaussian integration-by-parts step in Lemma 3.1 uses $\sigma^{-1}$; testing whether the Wasserstein rate survives under weaker nondegeneracy would separate the two metrics' requirements.
The same domino decomposition with an integration-by-parts gradient estimate should transfer to other explicit locally Lipschitz schemes, such as truncated, adaptive, or implicit Euler–Maruyama methods, giving uniform-in-time distributional rates under the same dissipativity and ellipticity conditions.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: this is a genuinely new uniform-in-time W1/TV convergence result for a known explicit tamed scheme, and the proof is in good shape; the real cost is a restrictive but clearly stated uniform ellipticity/boundedness assumption on the diffusion.

What is new: the tamed EM scheme (1.2) is Sabanis's, but the infinite-horizon analysis with decreasing step sizes is new. Theorems 1.1 and 1.2 give W1 and TV rates O(η_n^α) for any α∈(0,1/2), uniformly in n, under partial dissipation. That is exactly the gap left by the finite-time Lp result [13] and the divergence result [8]. The proof is a domino decomposition plus Bismut-Elworthy-Li semigroup estimates plus careful summation (Lemma 3.2); all the pieces fit. I read the proof and found no load-bearing error. The stress-test note is right: A2 is a scope restriction, not a hidden gap.

Soft spots: (1) A2 is strong: sigma globally bounded with bounded derivatives and sigma^{-1} uniformly bounded. Degenerate or growing noise is excluded. That should be front and center. (2) The rate is η_n^α with α<1/2, and the same α appears in the taming; it does not reach the usual finite-time order 1/2. Fine, but calibrate expectations. (3) Minor: in Lemma 2.3 the application of Lemma 2.2 to the scaled variable 3ξ is not fully spelled out; the implied condition η‖Σ‖≤1/54 holds for η small but should be stated. The passage from C^2_b test functions to Lip(1)/B_b is routine but terse. (4) No numerical illustrations; this is a pure proof paper, so no data to check. The citation pattern is healthy: no self-citations, and the contrast with [13], [8], and [7] is accurate.

Who this is for: anyone analyzing explicit schemes for superlinear SDEs over long horizons, especially for sampling/ergodic applications. I would send it to a serious referee and expect acceptance after minor revisions.

Referee Report

0 major / 5 minor

Summary. The paper proves uniform-in-time distributional approximation rates for a tamed Euler-Maruyama scheme (1.2) applied to the SDE (1.1) with polynomially growing, dissipative drift and multiplicative noise. Under Assumptions A1-A3 (dissipativity and polynomial growth of b; bounded, uniformly elliptic C^2 diffusion coefficient sigma; non-increasing step sizes satisfying eta_{n-1}-eta_n <= theta eta_n^2), Theorem 1.1 gives W1 and total-variation error bounds of order eta_n^alpha for every alpha in (0,1/2), uniformly in n, and Theorem 1.2 states the additive-noise counterpart. The proof combines a domino decomposition (3.1), one-step error estimates (Lemma 2.4), uniform moment bounds for X and Y (Lemmas 2.1 and 2.3), Bismut-Elworthy-Li gradient estimates for the semigroup of X (Lemma 2.8), and summation estimates for variable step sequences (Lemma 3.2).

Significance. If correct, the result supplies a long-time, uniform-in-time convergence theory for an explicit EM-type scheme in the multiplicative-noise setting, including a total-variation rate, which is considerably stronger than typical finite-time L^p convergence results. The proof is detailed and largely self-contained, with the main estimates assembled cleanly: the rate is stated as an explicit function of the step-size sequence, and the dependence of constants on the problem data is tracked. The principal limitation is the strength of Assumption A2, which globally bounds sigma and requires uniform ellipticity; this excludes degenerate or growing diffusions, but it is explicitly stated and is used essentially in the semigroup gradient estimates and the Gaussian integration-by-parts argument.

minor comments (5)

[Lemma 2.3] In the proof around (2.8), the decomposition of U(Y_{t_n}) uses the ball B(mu,1/9), while Lemma 2.2 is then applied with B(mu,1/3). Please clarify that Lemma 2.2 is applied to the scaled variable 3Y_{t_n}; after scaling, the relevant ball is B(3mu,1/3) and the smallness condition becomes eta_n ||Sigma|| <= 1/54, not merely 1/6. The current text is confusing and the radius mismatch should be corrected.
[Lemma 3.2] The condition in the statement reads 'theta < c e^{-c/beta}', which is ambiguous. The proof requires theta < c e^{-c}/beta, that is, beta appears in the denominator of the bound, not in the exponent. Please restate the hypothesis precisely.
[Proof of Theorem 1.1, after (3.6)] The absorption of eta_n^{1/2}|ln eta_n| into eta_n^alpha uses eta_n <= 1 and alpha < 1/2. It would be helpful to state explicitly that finitely many early indices, for which eta_n is not small, are absorbed into the constant C.
[Section 3, beginning] The 'standard approximation method' that passes from the C^2_b test-function estimate (3.8) to the W1 and total-variation bounds is only mentioned in one sentence. A short explanation of how bounded measurable and Lipschitz test functions are approximated by C^2_b functions with controlled norms would make the argument fully self-contained.
[Throughout] There are several typographical and OCR-style artifacts, such as 'TAMED EULER-MARUY AMA' in the running title, 'F or ad-dimensional' at the start of Lemma 2.2, and assorted broken ligatures in the displayed text. A careful copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence rates are derived from stated assumptions, not from the target quantities.

full rationale

The paper derives its uniform-in-time Wasserstein and total variation convergence rates from explicit hypotheses A1–A3 and from standard external tools (Bismut–Elworthy–Li formula, exponential ergodicity of the continuous semigroup, and a Girsanov-based strong Feller/irreducibility argument). The constant alpha in the rate is a freely chosen taming exponent in (0,1/2), not a fitted parameter; the final eta_n^alpha bound is obtained by absorbing the logarithmic factor produced by Lemma 3.2(iii) into eta_n^{1/2}|ln eta_n| <= C eta_n^alpha. There are no self-citations to the present authors' prior work, no equation is defined in terms of the quantity being proved, and no fitted quantity is renamed as a prediction. The restrictive uniform ellipticity condition in A2 is an explicit modeling assumption used in the gradient estimates; it limits scope but does not make the argument circular.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No fitted parameters and no invented entities. The paper's central result is a conditional derivation: given assumptions A1-A3 and standard theorems of stochastic analysis, it proves a convergence rate. The single user-chosen parameter alpha controls the taming exponent; it is not fitted to data. External theorems used are classical.

free parameters (1)

alpha = any alpha in (0,1/2)
Exponent in the taming factor 1/(1+eta_n^alpha ||nabla b||) and in the convergence rate O(eta_n^alpha). It is a user-chosen method parameter, not fitted to data; the theorem holds for every value in the interval.

assumptions (5)

standard math Existence and uniqueness of a strong solution to SDE (1.1) under A1-A2, cited to Prevot-Rockner, Theorem 3.3.1.
Invoked at the end of Section 1.2 to define X_t.
standard math Bismut-Elworthy-Li formula (Lemma 2.5) for first-order derivatives of the heat semigroup.
Used throughout Lemma 2.8 to obtain gradient estimates; proof cited to [1,3].
standard math Exponential ergodicity contraction with polynomial weight, [7, Theorem 2.5(a)].
Used in Lemma 2.8 for t >= 1 to bound the semigroup oscillation; the paper verifies strong Feller and irreducibility itself in Lemma 2.7.
standard math Girsanov theorem for change of measure.
Used in Lemma 2.7 to construct a process with the same law as X_t that lands in a prescribed ball with positive probability.
standard math Density of smooth bounded functions in approximation of bounded measurable functions, cited to Folland, Theorems 7.10 and 8.14.
Used in Lemma 2.7 to extend strong Feller from C_b^1 to B_b.

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Cite this review

Pith. "Pith review of Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise." pith.science (2026). https://pith.science/paper/N7HL6T6O

@misc{pith2026250503104,
  author       = {Pith},
  title        = {Pith review of: Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7HL6T6O}},
  note         = {Machine review of arXiv:2505.03104}
}

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 + \sigma (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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [13]

Sotirios Sabanis, Euler approximations with varying coefﬁcients: The case of superlinearly growing dif- fusion coefﬁcients , Annals of Applied Probability 26 (2013), 2083–2105. 1, 2

work page 2013

[2] [8]

2130, 1563–1576

Martin Hutzenthaler, Arnulf Jentzen, and Peter E Kloeden , Strong and weak divergence in ﬁnite time of euler’s method for stochastic differential equations with non-globally lipschitz continuous coefﬁcients , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467 (2011), no. 2130, 1563–1576. 1

work page 2011

[3] [7]

Notes Pure Appl

Beniamin Goldys and Bohdan Maslowski, Exponential ergodicity for stochastic reaction-diffusio n equa- tions, Stochastic partial differential equations and applicati ons–VII, Lect. Notes Pure Appl. Math., vol. 245, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 115–131. M R 2227225 13, 16

work page 2006

[4] [1]

Jean-Michel Bismut, Large deviations and the malliavin calculus , Birkhauser Prog. Math. 45 (1984). 9

work page 1984

[5] [2]

Minh-Thang Do, Hoang-Long Ngo, and Nhat-An Pho, T amed-adaptive euler-maruyama approximation for sdes with superlinearly growing and piecewise continuo us drift, superlinearly growing and locally hölder continuous diffusion, Journal of Complexity 82 (2024), 101833. 1

work page 2024

[6] [3]

1, 252–286

K David Elworthy and Xue-Mei Li, F ormulae for the derivatives of heat semigroups, Journal of Functional Analysis 125 (1994), no. 1, 252–286. 9

work page 1994

[7] [4]

Wei Fang and Michael B Giles, Adaptive euler-maruyama method for sdes with non-globally lipschitz drift: Part ii, inﬁnite time interval , arXiv preprint arXiv:1703.06743 (2017). 1

work page Pith review arXiv 2017

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Gerald B Folland, Real analysis: modern techniques and their applications , vol. 40, John Wiley & Sons,

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