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arxiv: 2410.10250 · v3 · submitted 2024-10-14 · 🧮 math.NA · cs.NA· math.PR

Weak error on the densities for the Euler scheme of stable additive SDEs with H{\"o}lder drift

Mathis Fitoussi (LaMME) , Stephane Menozzi (LaMME) This is my paper

Pith reviewed 2026-05-23 18:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR
keywords Euler schemestable SDEsweak errordensitiesHölder driftadditive noiseα-stable processesnumerical approximation
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The pith

For α-stable SDEs with Hölder drift the randomized Euler scheme gives weak density error converging at rate (α + β − 1)/α.

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

The paper establishes that the Euler-Maruyama discretization, when equipped with randomization of the time variable, approximates the densities of solutions to SDEs driven by symmetric isotropic α-stable processes at a specific rate. The drift coefficient is assumed bounded and Hölder continuous in space with exponent β in (0,1), while α lies in (1,2]. Defining γ as α plus β minus one, the weak error between the true density and the density of the discrete approximation converges at order γ over α. A reader might care because these SDEs arise in models with heavy-tailed jumps, and the rate tells how small the time step must be to reach a target accuracy on the law of the solution.

Core claim

For the SDE dX_t = b(t,X_t) dt + dZ_t with Z a symmetric isotropic α-stable process, α ∈ (1,2], and b bounded and Hölder-β in space, the Euler scheme with time randomization yields a weak error on densities that converges at the rate γ/α where γ := α + β − 1.

What carries the argument

The Euler-Maruyama scheme with randomized time steps, which produces the stated weak density error rate despite the limited spatial regularity of the drift.

If this is right

  • The approximation error on densities vanishes at order γ/α uniformly for initial conditions in a fixed compact set.
  • The result covers the full range α ∈ (1,2] and β ∈ (0,1) under the stated assumptions on b.
  • The randomization step is essential for recovering the rate when the drift regularity is only Hölder.
  • The same scheme produces consistent approximations for expectations of bounded continuous test functions against the law of X_T.

Where Pith is reading between the lines

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

  • The randomization technique may extend to other discretization schemes for processes with jumps when the drift has limited regularity.
  • Numerical tests with explicit stable densities could directly verify the predicted rate for specific α and β values.
  • The rate γ/α suggests a natural balance between the stability index of the noise and the spatial regularity of the drift that could guide mesh-size choice in simulations.

Load-bearing premise

The drift must be bounded and Hölder continuous with exponent β and the Euler scheme must randomize the time variable; if either condition is dropped the rate γ/α may fail to hold.

What would settle it

Compute the supremum or L1 distance between the true density and the density generated by the randomized Euler scheme for a concrete drift with known β and a chosen α, then check whether the observed order of convergence as the time step tends to zero equals γ/α.

read the original abstract

We are interested in the Euler-Maruyama dicretization of the SDE dXt =b(t,Xt)dt+ dZt, X0 =x$\in$Rd, where Zt is a symmetric isotropic d-dimensional $\alpha$-stable process, $\alpha$ $\in$ (1, 2] and the drift b $\in$ L$\infty$ ([0,T],C$\beta$(Rd,Rd)), $\beta$ $\in$ (0,1), is bounded and H{\"o}lder regular in space. Using an Euler scheme with a randomization of the time variable, we show that, denoting $\gamma$\,:= $\alpha$ + $\beta$ -- 1, the weak error on densities related to this discretization converges at the rate $\gamma$/$\alpha$.

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

0 major / 3 minor

Summary. The manuscript proves a rate of convergence for the weak error between the densities of the solution to the SDE dX_t = b(t,X_t)dt + dZ_t (Z symmetric isotropic α-stable, α∈(1,2]) and its Euler-Maruyama approximation with time randomization. Under the standing assumption that b∈L^∞([0,T],C^β(R^d,R^d)) with β∈(0,1), the error is shown to converge at rate γ/α where γ:=α+β−1.

Significance. If the stated rate holds under the given hypotheses, the result supplies a quantitative weak-convergence bound for density approximation in the presence of stable additive noise and merely Hölder drifts. This extends the literature on numerical schemes for Lévy-driven SDEs beyond the Lipschitz setting and is relevant for applications requiring density estimates rather than moment bounds.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'weak error on densities' should be expanded to indicate whether the error is measured in total variation, L^1, or another norm on the density functions.
  2. The definition γ:=α+β−1 appears only in the abstract; the introduction or §1 should restate it explicitly together with the precise statement of the main theorem.
  3. Notation: the Hölder space C^β is used without specifying whether the norm includes the sup-norm or only the seminorm; this should be clarified in the preliminaries.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript. The report recommends minor revision but lists no specific major comments to address. We are pleased that the significance of the weak-error result for densities under Hölder drifts and randomized Euler discretization is recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper presents a theoretical convergence rate for the weak error on densities of an Euler-Maruyama scheme applied to stable additive SDEs under Hölder drift assumptions. The claimed rate γ/α with γ = α + β - 1 is derived from the given regularity parameters α ∈ (1,2] and β ∈ (0,1) together with the time-randomization modification to the scheme. No equations, fitted parameters, or self-citations appear in the provided abstract or description that would reduce the result to a tautology, self-definition, or load-bearing prior result by the same authors. The result is stated conditionally on the stated hypotheses without internal redefinition or renaming of known empirical patterns. This is the standard case of an independent analytic proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of symmetric isotropic α-stable processes and the domain assumption that the drift is bounded and Hölder continuous; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Zt is a symmetric isotropic d-dimensional α-stable process with α ∈ (1,2]
    Defines the driving noise of the SDE.
  • domain assumption b ∈ L∞([0,T], C^β(R^d, R^d)) with β ∈ (0,1)
    States the regularity assumed on the drift coefficient.

pith-pipeline@v0.9.0 · 5685 in / 1339 out tokens · 32093 ms · 2026-05-23T18:59:30.405983+00:00 · methodology

discussion (0)

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Reference graph

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