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arxiv: 1907.02293 · v1 · pith:NSSJUOXVnew · submitted 2019-07-04 · 🧮 math.PR

Weak convergence of path-dependent SDEs driven by fractional Brownian motion with irregular coefficients

Yongqiang Suo , Chenggui Yuan , shaoqin Zhang This is my paper

Pith reviewed 2026-05-25 09:24 UTC · model grok-4.3

classification 🧮 math.PR
keywords fractional Brownian motionstochastic functional differential equationsweak existence and uniquenessEuler-Maruyama schemeGirsanov transformationHölder continuous driftHurst indexweak convergence
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The pith

Girsanov transformation establishes weak existence, uniqueness and Euler-Maruyama convergence for path-dependent SDEs with Hölder drift under fractional Brownian motion.

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

The paper establishes weak existence and uniqueness in law for stochastic functional differential equations whose drift is merely Hölder continuous and whose driving noise is fractional Brownian motion with Hurst index in (1/2,1). It further shows that the Euler-Maruyama discretisation converges weakly to any such solution. The argument proceeds by a change of measure that reduces the original equation to a reference equation whose properties can be transferred back. A reader cares because the result covers equations with memory and with coefficients too irregular for classical Lipschitz theory.

Core claim

Using Girsanov's transformation together with known properties of the corresponding reference stochastic differential equations, the authors prove that stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with H ∈ (1/2,1) possess weak solutions that are unique in law and that the Euler-Maruyama scheme converges weakly to these solutions.

What carries the argument

Girsanov's transformation applied to reference SDEs, which transfers weak existence, uniqueness and approximation properties from the reference equations back to the original path-dependent equations.

If this is right

  • Weak solutions exist for these equations even when the drift fails to be Lipschitz.
  • The Euler-Maruyama scheme converges weakly whenever the reference equation satisfies the transferred conditions.
  • The results apply uniformly for all Hurst indices strictly between 1/2 and 1.
  • Path dependence can be accommodated without losing the weak convergence conclusion.

Where Pith is reading between the lines

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

  • The same change-of-measure device may extend to other Gaussian drivers that admit a Girsanov-type theorem.
  • Numerical schemes beyond Euler-Maruyama could be analysed by verifying the corresponding reference properties.
  • Applications in rough-volatility or long-memory models become justifiable once the reference SDE properties are checked.

Load-bearing premise

The reference SDEs obtained after the Girsanov change of measure must have weak existence, uniqueness and approximation properties that carry over to the original path-dependent equation.

What would settle it

An explicit example of a Hölder-continuous drift and fractional Brownian motion path for which the transformed reference equation fails to have unique weak solutions or for which the Euler-Maruyama scheme diverges in law from the true solution.

read the original abstract

In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with H\"older continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.

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 / 1 minor

Summary. The manuscript claims that weak existence and uniqueness of solutions, together with weak convergence of the Euler-Maruyama scheme, hold for stochastic functional differential equations with Hölder-continuous drift driven by fractional Brownian motion (H ∈ (1/2,1)). The strategy consists of applying Girsanov's transformation to reduce the original path-dependent equation to reference SDEs whose weak well-posedness and scheme convergence are taken as known.

Significance. If the Girsanov step is rigorously justified, the result would modestly extend existing weak-convergence theory from ordinary SDEs to the functional setting with low-regularity coefficients. The reduction via Girsanov is a standard technique, but its successful application here would require explicit control on the Cameron-Martin regularity of the transformed drift.

major comments (2)
  1. [Abstract and §1] Abstract and §1 (strategy outline): the reduction via Girsanov presupposes that the integrated drift process lies in the Cameron-Martin space of the fBM almost surely. For a Hölder-α coefficient the solution regularity is at most Hölder-γ with γ < H; the L²-integrability condition after application of K_H^{-1} therefore requires an explicit lower bound α > 1-H that is neither stated nor verified. This assumption is load-bearing for the entire argument.
  2. [Abstract] The manuscript invokes unspecified 'properties of the corresponding reference stochastic differential equations' without identifying the precise theorems being transferred or confirming that the path-dependent structure does not interfere with those properties after the Girsanov change of measure.
minor comments (1)
  1. Notation for the fractional kernel operator K_H and its inverse should be introduced with a brief recall of the relevant Cameron-Martin norm before the transformation is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and will make the necessary revisions to clarify the assumptions and references.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (strategy outline): the reduction via Girsanov presupposes that the integrated drift process lies in the Cameron-Martin space of the fBM almost surely. For a Hölder-α coefficient the solution regularity is at most Hölder-γ with γ < H; the L²-integrability condition after application of K_H^{-1} therefore requires an explicit lower bound α > 1-H that is neither stated nor verified. This assumption is load-bearing for the entire argument.

    Authors: We agree that an explicit condition α > 1-H is required to guarantee that the integrated drift process belongs to the Cameron-Martin space a.s. and satisfies the necessary L²-integrability after applying K_H^{-1}. This was implicitly used in our Girsanov argument but not stated. In the revision we will add the assumption α > 1-H to the hypotheses on the drift and include a short verification that the resulting process satisfies the Cameron-Martin condition. revision: yes

  2. Referee: [Abstract] The manuscript invokes unspecified 'properties of the corresponding reference stochastic differential equations' without identifying the precise theorems being transferred or confirming that the path-dependent structure does not interfere with those properties after the Girsanov change of measure.

    Authors: The reference equations are the standard driftless SDEs driven by fBM (with continuous coefficients). Weak well-posedness and Euler-Maruyama weak convergence for these equations are taken from known results in the literature on fBM-driven SDEs. The Girsanov transformation removes the (path-dependent) drift term, so the reference equation under the new measure is non-functional. We will cite the precise theorems used and add a sentence in the abstract and §1 confirming that the path-dependent structure is eliminated by the change of measure. revision: yes

Circularity Check

0 steps flagged

No circularity: Girsanov reduction to reference SDEs is independent of target result

full rationale

The paper applies Girsanov transformation to map the original path-dependent SFDE (Hölder drift, fBM driver) onto reference SDEs, then pulls back weak existence/uniqueness and EM convergence from those references. No equation or step defines the target result in terms of itself, renames a fitted quantity as a prediction, or relies on a self-citation chain whose content is unverified. The reference properties are treated as external inputs; the transfer step does not presuppose the final theorem. This matches the default case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on the unelaborated 'property of the corresponding reference stochastic differential equations' after Girsanov transformation.

pith-pipeline@v0.9.0 · 5574 in / 1206 out tokens · 30696 ms · 2026-05-25T09:24:12.722225+00:00 · methodology

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