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arxiv: 2605.16900 · v1 · pith:25ZR7DUNnew · submitted 2026-05-16 · 📊 stat.ME · math.ST· stat.TH

Splitting schemes and estimators for stochastic differential equations with H\"older multiplicative noise

Bowen Fang , Dario Span\`o , Massimiliano Tamborrino This is my paper

Pith reviewed 2026-05-19 20:09 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords stochastic differential equationsparameter estimationsplitting schemesLamperti transformpseudo-likelihood estimatorsHölder continuous diffusionstrong mean-square convergenceasymptotic normality
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The pith

Splitting schemes based on the Lamperti transform produce strongly convergent and state-space-preserving pseudo-likelihood estimators for SDEs with Hölder multiplicative noise.

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

This paper addresses parameter estimation for univariate stochastic differential equations with locally Lipschitz drift and Hölder continuous multiplicative diffusion. Standard Euler-Maruyama discretizations typically lack strong convergence and do not preserve the state space, while alternative approximations reduce stability. The authors derive explicit pseudo-likelihood estimators from Lie-Trotter and Strang splitting schemes applied after a Lamperti transform that decomposes the equation into drift and diffusion parts. They prove strong mean-square convergence and state-space preservation for these schemes, along with improved robustness to step size, and show consistency and asymptotic normality for the Lie-Trotter estimator via new techniques that handle parameter coupling. Readers care because these SDEs arise in applications yet have lacked reliable explicit inference methods.

Core claim

We introduce the first explicit pseudo-likelihood estimators based on numerical splitting schemes that are both strong mean-square convergent and state space preserving for this class of SDEs. Our approach is based on a novel decomposition of the SDE that exploits reducibility and the Lamperti transform, leading to Lie-Trotter (LT) and Strang splitting schemes yielding explicit pseudo-likelihoods and maximum likelihood estimators based on them. We prove strong mean-square convergence, state space preservation, and improved robustness with respect to the discretisation step compared to Euler-Maruyama-based methods. We further establish consistency and asymptotic normality of the LT estimator.

What carries the argument

Lamperti transform to reduce multiplicative noise to additive form, followed by Lie-Trotter and Strang splitting to explicitly separate and simulate drift and diffusion for pseudo-likelihood construction.

If this is right

  • The Lie-Trotter estimator is consistent and asymptotically normal, enabling reliable confidence intervals and inference.
  • State space preservation by the schemes supports modeling of processes that must remain in a restricted domain such as positive values.
  • Improved robustness to discretization step size permits larger steps without loss of accuracy compared to Euler-Maruyama.
  • Simulations confirm higher accuracy and lower computational cost than existing approximation-based estimators.
  • The new asymptotic techniques handle the coupling between drift and diffusion parameters in the pseudo-likelihood.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Lamperti-plus-splitting construction may extend to other reducible SDE classes if the transform preserves the required regularity.
  • Long-horizon simulations in applications could become cheaper by using larger stable time steps while retaining convergence.
  • Multivariate versions would be a direct test if analogous reducibility can be established for vector noise.
  • Empirical studies on real data sets from finance or biology could quantify gains in estimate stability over current practice.

Load-bearing premise

The SDE admits a reducible decomposition via the Lamperti transform that allows explicit splitting into drift and diffusion components while preserving the Hölder regularity and local Lipschitz conditions needed for the convergence and likelihood derivations.

What would settle it

A simulation on a concrete test SDE with Hölder multiplicative diffusion in which the mean-square error of trajectories from the splitting scheme fails to decrease proportionally to the time step size would falsify the strong convergence result.

Figures

Figures reproduced from arXiv: 2605.16900 by Bowen Fang, Dario Span\`o, Massimiliano Tamborrino.

Figure 1
Figure 1. Figure 1: Illustration of the mean-square convergence order on the CIR process via the [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Inference results for the CIR process: densities of the pseudo MLEs derived [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Inference results for the Student diffusion: densities of the pseudo MLEs derived [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Inference result for the Ahn-Gao model: violin plots of the estimates derived [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Inference results for the F diffusion: densities of the pseudo MLEs derived [PITH_FULL_IMAGE:figures/full_fig_p047_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ten trajectories of the Wright-Fisher simulated using the LT and S splitting [PITH_FULL_IMAGE:figures/full_fig_p048_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Wright-Fisher diffusion: Empirical distribution of the LT splitting scheme ( [PITH_FULL_IMAGE:figures/full_fig_p049_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the mean-square convergence order on the Student diffusion via [PITH_FULL_IMAGE:figures/full_fig_p050_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the mean-square convergence order on the F diffusion via the [PITH_FULL_IMAGE:figures/full_fig_p050_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the mean-square convergence order on the Wright-Fisher via [PITH_FULL_IMAGE:figures/full_fig_p051_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Wasserstein distance between the one-step transition density (from time 0 to [PITH_FULL_IMAGE:figures/full_fig_p052_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of the mean-square convergence order on the Ginzburg-Landau [PITH_FULL_IMAGE:figures/full_fig_p053_12.png] view at source ↗
read the original abstract

We study parameter estimation for univariate stochastic differential equations with locally Lipschitz drift and H\"older continuous multiplicative diffusion, a class commonly arising in several applications. Existing inference methods typically rely on either the Euler-Maruyama discretisation, despite its lack of strong convergence and failure to preserve the state space, or on approximations, e.g. Gaussian approximation or truncation of Hermite's expansions, impacting on their stability and computational efficiency. We introduce the first explicit pseudo-likelihood estimators based on numerical splitting schemes that are both strong mean-square convergent and state space preserving for this class of SDEs. Our approach is based on a novel decomposition of the SDE that exploits reducibility and the Lamperti transform, leading to Lie-Trotter (LT) and Strang splitting schemes yielding explicit pseudo-likelihoods and maximum likelihood estimators based on them. We prove strong mean-square convergence, state space preservation, and improved robustness with respect to the discretisation step compared to Euler-Maruyama-based methods. We further establish consistency and asymptotic normality of the LT estimator. Because the proposed numerical scheme couples drift and diffusion parameters in the pseudo-likelihood, the asymptotic analysis requires new proof techniques. Extensive simulations demonstrate that the proposed estimators outperform existing methods in both accuracy and computational efficiency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops explicit pseudo-likelihood estimators for univariate SDEs with locally Lipschitz drift and Hölder multiplicative diffusion. It exploits reducibility via the Lamperti transform to obtain an additive-noise equation, then constructs Lie-Trotter and Strang splitting schemes that yield closed-form transition densities for the pseudo-likelihood. The authors prove strong mean-square convergence and state-space preservation of the schemes, establish consistency and asymptotic normality of the Lie-Trotter estimator (requiring new techniques because drift and diffusion parameters are coupled), and report simulation evidence of improved accuracy and robustness relative to Euler-Maruyama-based estimators.

Significance. If the regularity and convergence claims hold, the work supplies the first explicit, strongly convergent, state-space-preserving pseudo-likelihood estimators for this important class of SDEs. The proofs of strong convergence, state-space preservation, consistency, and asymptotic normality, together with the explicit splitting-based likelihoods, constitute a substantive methodological advance for applications in which standard discretizations fail.

major comments (2)
  1. [§3] §3 (Lamperti decomposition): the argument that the transformed drift remains locally Lipschitz (or satisfies the precise regularity needed for the splitting convergence theory) when the original diffusion coefficient is merely α-Hölder with α<1 is not fully detailed. The Itô correction term arising from the Lamperti map can have a local Lipschitz constant controlled by the Hölder modulus of σ; explicit bounds or counter-example checks are required to confirm that the conditions invoked for the Lie-Trotter/Strang convergence theorems remain satisfied on the relevant state-space regions.
  2. [§5] §5 (asymptotic normality of the LT estimator): the proof must rigorously control the discretization bias induced by the splitting scheme inside the coupled pseudo-likelihood. Because the estimator jointly depends on drift and diffusion parameters, the usual martingale or ergodic arguments need to be adapted; the manuscript should state the precise rate at which the splitting step-size h_n → 0 relative to the sample size n so that the asymptotic normality statement holds.
minor comments (2)
  1. [Simulations] Table 1 and Figure 2: the reported MSE values for the Strang scheme appear sensitive to the choice of initial step-size; add a brief sensitivity plot or table entry for h = 0.01, 0.05, 0.1 to illustrate the claimed robustness.
  2. [Notation] Notation: the Hölder exponent α is introduced in the abstract and §2 but is not carried explicitly into the statements of Theorems 3.1 and 4.2; make the dependence on α visible in the convergence rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the major comments point by point below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: §3 (Lamperti decomposition): the argument that the transformed drift remains locally Lipschitz (or satisfies the precise regularity needed for the splitting convergence theory) when the original diffusion coefficient is merely α-Hölder with α<1 is not fully detailed. The Itô correction term arising from the Lamperti map can have a local Lipschitz constant controlled by the Hölder modulus of σ; explicit bounds or counter-example checks are required to confirm that the conditions invoked for the Lie-Trotter/Strang convergence theorems remain satisfied on the relevant state-space regions.

    Authors: We acknowledge that the details on the regularity of the transformed drift could be expanded for clarity. The Lamperti transform applied to the multiplicative noise SDE yields an additive noise equation where the new drift includes an Itô correction term involving the derivative of the inverse transform and the diffusion coefficient. Since σ is α-Hölder continuous with α < 1, its derivative may not exist, but the correction term's Lipschitz property can be established locally using the Hölder modulus. In the revised manuscript, we will include an additional lemma that provides explicit bounds showing that the local Lipschitz constant of the transformed drift is bounded by a constant depending on the Hölder constant of σ and the distance to the boundary of the state space. This will verify that the assumptions for the strong convergence of the splitting schemes hold. revision: yes

  2. Referee: §5 (asymptotic normality of the LT estimator): the proof must rigorously control the discretization bias induced by the splitting scheme inside the coupled pseudo-likelihood. Because the estimator jointly depends on drift and diffusion parameters, the usual martingale or ergodic arguments need to be adapted; the manuscript should state the precise rate at which the splitting step-size h_n → 0 relative to the sample size n so that the asymptotic normality statement holds.

    Authors: We agree that specifying the rate is important for the asymptotic result. Our current proof adapts ergodic theorems and martingale central limit theorems to handle the coupling between drift and diffusion parameters in the pseudo-likelihood. The discretization bias from the Lie-Trotter scheme is controlled by the strong convergence order of the scheme. To address the referee's point, we will revise the statement of the asymptotic normality theorem to explicitly require that the step-size satisfies h_n = o(n^{-1/2}), ensuring the bias term vanishes in the limit. We will also add a brief explanation of how the bias is bounded in the proof. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with independent proofs and novel techniques

full rationale

The paper introduces explicit pseudo-likelihood estimators for SDEs with locally Lipschitz drift and Hölder multiplicative diffusion by applying the Lamperti transform to obtain a reducible decomposition, then constructing Lie-Trotter and Strang splitting schemes. It proves strong mean-square convergence, state-space preservation, consistency, and asymptotic normality using new proof techniques necessitated by the coupling of drift and diffusion parameters in the pseudo-likelihood. These steps are derived from stated assumptions on the SDE class and do not reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central results rest on independent derivations rather than tautological equivalences with the estimator definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on the existence of a Lamperti transform that reduces the multiplicative noise SDE to an additive form while preserving the required regularity; no free parameters are introduced beyond standard discretization step size, and no new entities are postulated.

axioms (1)
  • domain assumption The SDE admits a Lamperti transform that yields an equivalent additive-noise equation under the given local Lipschitz drift and Hölder diffusion conditions.
    Invoked to enable the decomposition into drift and diffusion components for splitting (abstract).

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