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arxiv: 2510.25151 · v2 · submitted 2025-10-29 · 🧮 math.PR

L^(α-1) distance between two one-dimensional stochastic differential equations with drift terms driven by a symmetric α-stable process

Takuya Nakagawa This is my paper

Pith reviewed 2026-05-18 03:47 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic differential equationssymmetric alpha-stable processesstability theoryHolder estimatespathwise comparisontime-dependent coefficientstransition densities
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The pith

The L^{α-1} distance between solutions of one-dimensional α-stable SDEs with drifts satisfies a Hölder-type bound controlled by initial differences and a weighted coefficient norm.

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

This paper aims to prove a quantitative stability estimate for one-dimensional stochastic differential equations driven by symmetric α-stable processes that include nonzero drift terms and time-dependent coefficients. It establishes that the L^{α-1} distance between two such solution paths obeys a Hölder continuity bound with respect to the difference in initial conditions and a novel weighted integral norm measuring the difference in their coefficients. The weighted norm is derived from the transition probability density of a baseline solution to handle perturbations that vary with time. A reader would care if this holds because it supplies the first explicit rates for how solutions respond to changes in their driving terms, enabling better analysis of approximations and perturbations in these jump-driven processes.

Core claim

The paper claims that for two solutions X and Y to SDEs of the form dX_t = b(t, X_t) dt + dZ_t and dY_t = c(t, Y_t) dt + dZ_t where Z is a symmetric α-stable process with 1 < α < 2, the L^{α-1}(Ω) distance between solution paths satisfies a Hölder-type estimate that quantifies stability with respect to initial values and coefficients, with the coefficient difference measured in a weighted integral norm built from the transition probability density of the baseline solution rather than a supremum norm.

What carries the argument

The weighted integral norm for the difference between coefficients, constructed from the transition probability density of the baseline solution, which localizes the error analysis and accommodates time-dependent perturbations.

If this is right

  • The stability estimate yields explicit convergence rates for the solutions when the coefficients converge in this weighted norm.
  • Corresponding bounds are derived for the probability that the uniform distance between paths exceeds a given level.
  • The refined analysis of the mollified auxiliary function provides sharper control over the effects of nonzero drift terms.
  • The method generalizes the pathwise comparison technique to stability problems with time-dependent coefficients.

Where Pith is reading between the lines

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

  • Similar weighted norms could be applied to derive stability results for SDEs driven by other Lévy processes with accessible transition densities.
  • The approach may extend to provide error estimates for numerical discretization schemes applied to these equations.
  • Quantitative stability of this form could support sensitivity analysis in models using α-stable drivers.

Load-bearing premise

The transition probability density of the baseline solution exists and can be used to define the weighted integral norm that localizes the error analysis.

What would settle it

A specific pair of SDEs with small initial difference and small coefficient difference in the weighted norm but with L^{α-1} distance between paths larger than any Hölder bound would disprove the estimate.

read the original abstract

This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $\alpha$-stable process for $\alpha\in(1,2)$. Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially, time-dependent coefficients. We provide the first explicit convergence rates for this broad class of SDEs. The main result is a H\"older-type estimate for the $L^{\alpha-1}(\Omega)$ distance between two solution paths, which quantifies the stability with respect to the initial values and coefficients. A key innovation of our approach is the measurement of the distance between coefficients. Instead of using a standard supremum norm, which would impose restrictive conditions, we introduce a weighted integral norm constructed from the transition probability density of the baseline solution. This technique, which generalizes the framework of Nakagawa \cite{Nakagawa}, is essential for handling time-dependent perturbations and effectively localizes the error analysis. The proof is based on a refined analysis of a mollified auxiliary function, for which we establish a new, sharper derivative estimate to control the drift terms. Finally, we apply these stability results to derive corresponding convergence rates in probability, providing an upper bound for the tail probability of the uniform distance between solution paths.

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 establishes a quantitative stability result for one-dimensional SDEs driven by symmetric α-stable processes (α ∈ (1,2)) with non-zero, time-dependent drifts. It proves a Hölder-type bound on the L^{α-1}(Ω) distance between two solution paths that quantifies stability with respect to initial conditions and coefficient perturbations. The key technical device is a weighted integral norm on coefficient differences, constructed from the transition density of a baseline solution, together with a refined mollified auxiliary function that yields sharper derivative estimates to control the drifts. The result generalizes pathwise comparison methods of Komatsu and Nakagawa and is applied to obtain convergence rates in probability.

Significance. If the central estimates hold, the work supplies the first explicit convergence rates for this class of SDEs with time-dependent coefficients, which is a meaningful advance in the stability theory of stable-driven equations. The introduction of the weighted norm (instead of a plain sup-norm) and the sharper derivative bounds on the mollified test function are concrete technical contributions that could be useful beyond the present setting.

major comments (2)
  1. The weighted integral norm is defined using the transition density p(t,x,y) of the baseline solution. For symmetric α-stable drivers with non-zero drift, existence, Hölder regularity, and strict positivity of p are not automatic and typically require at least local Lipschitz or Hölder conditions on the drift together with non-degeneracy. The manuscript must explicitly state and verify these assumptions (or prove the requisite density bounds) before the weighted norm can be guaranteed to localize time-dependent perturbations and dominate the error term in the comparison argument.
  2. The sharper derivative estimates for the mollified auxiliary function are central to controlling the drift terms. The proof sketch in the abstract indicates these estimates are new, but the precise dependence on the mollification parameter and on the α-stable generator must be checked to confirm they remain uniform under the stated coefficient assumptions.
minor comments (2)
  1. The abstract cites Nakagawa but does not clarify which specific result is being generalized; a brief comparison paragraph in the introduction would help readers locate the novelty.
  2. Notation for the weighted norm (e.g., how the integral is taken with respect to the measure induced by p) should be introduced with an explicit formula early in the paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We sincerely thank the referee for the thorough review and valuable suggestions. We address the major comments point by point below, and we will incorporate the necessary clarifications and details in the revised version of the manuscript.

read point-by-point responses

  1. Referee: The weighted integral norm is defined using the transition density p(t,x,y) of the baseline solution. For symmetric α-stable drivers with non-zero drift, existence, Hölder regularity, and strict positivity of p are not automatic and typically require at least local Lipschitz or Hölder conditions on the drift together with non-degeneracy. The manuscript must explicitly state and verify these assumptions (or prove the requisite density bounds) before the weighted norm can be guaranteed to localize time-dependent perturbations and dominate the error term in the comparison argument.

    Authors: We agree with the referee that explicit assumptions and verification for the transition density are necessary. Although the existence of strong solutions implies some regularity, we will add in the revised manuscript a clear statement of the conditions on the drift coefficients (local Lipschitz continuity and boundedness) that ensure the transition density p(t,x,y) exists, is strictly positive, and satisfies Hölder regularity. We will also include a brief justification or reference to standard results for symmetric α-stable processes with drifts to confirm that the weighted integral norm localizes the perturbations effectively. revision: yes

  2. Referee: The sharper derivative estimates for the mollified auxiliary function are central to controlling the drift terms. The proof sketch in the abstract indicates these estimates are new, but the precise dependence on the mollification parameter and on the α-stable generator must be checked to confirm they remain uniform under the stated coefficient assumptions.

    Authors: We appreciate this observation. The sharper estimates are indeed derived in the proof using the generator of the α-stable process and the mollification. In the revised version, we will make the dependence on the mollification parameter explicit (e.g., bounds of order ε to a negative power depending on α) and prove that the constants remain uniform under the boundedness and continuity assumptions on the time-dependent drifts. This will be done by carefully tracking the constants in the estimates for the auxiliary function. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to Nakagawa framework; central Hölder estimate uses independent mollified estimates

full rationale

The paper's derivation of the Hölder-type L^{α-1}(Ω) path distance estimate proceeds via a new weighted integral norm on coefficient differences (built from the baseline transition density) together with sharper derivative bounds on a mollified auxiliary function. While the abstract notes that this norm generalizes the framework of Nakagawa (self-citation), the load-bearing steps are the fresh analytic estimates for the mollified function and the localization argument for time-dependent drifts; these do not reduce by construction to the cited prior work or to any fitted parameter. No self-definitional loops, predictions forced by input fits, or uniqueness theorems imported from the same authors appear in the provided chain. The result is therefore self-contained against external benchmarks for stable-process SDE stability.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard existence/uniqueness for the SDEs and on the existence of transition densities for the baseline process; the weighted norm is a technical construction rather than a new postulated entity.

axioms (2)
  • domain assumption Existence and uniqueness of strong solutions to the SDEs under the stated coefficient conditions
    Required for the distance between solutions to be well-defined.
  • domain assumption Existence and regularity of the transition probability density of the baseline solution
    Used to construct the weighted integral norm that localizes the error.

pith-pipeline@v0.9.0 · 5786 in / 1381 out tokens · 39537 ms · 2026-05-18T03:47:04.721733+00:00 · methodology

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