Stochastic Differential Equations with Discontinuous Diffusions

Lauri Viitasaari; Soledad Torres

REVIEW 3 major objections 4 minor 17 references

Stochastic Differential Equations with Discontinuous Diffusions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For one-dimensional SDEs driven by Hölder noises, discontinuous diffusion coefficients still yield explicit solutions via a time change.

desk verdict A promising method for discontinuous diffusions with Hölder drivers, but the main existence theorem is false as stated because it ignores the range of the Lamperti transform. read the letter →

arxiv 1908.03183 v1 pith:G7Z2FGIH submitted 2019-08-08 math.PR

Soledad Torres , Lauri Viitasaari This is my paper · ORCID

classification math.PR MSC 65C3060H0560G2226A33

keywords stochasticdifferentialequationfractionalcalculusHöldercontinuitydiscontinuityboundedvariationBrownianmotionpathwiseintegrationRosenblattprocess

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

The paper establishes that one-dimensional equations $dX_t=\sigma(X_t)\,dY_t$, driven by a Hölder-continuous process $Y$ of order $\alpha>1/2$, admit solutions even when $\sigma$ is discontinuous. The assumptions on $\sigma$ are mild: it must keep one sign, be of locally bounded variation, and have a locally integrable reciprocal. The solution is explicit: $X_t=\Lambda^{-1}(\Lambda(X_0)+Y_t-Y_0)$, where $\Lambda(x)=\int_a^x \frac{dy}{\sigma(y)}$. Uniqueness is proved up to the first time $\tau$ at which $\sigma(X_t)=0$, within a class of processes satisfying a sufficient-variability condition on the driver. This matters because it extends the classical Lipschitz theory of SDEs to jump-type diffusion coefficients and to drivers such as fractional Brownian motion with $H>1/2$ and the Rosenblatt process.

What carries the argument

The central object is the increasing map $\Lambda(x)=\int_a^x \frac{dy}{\sigma(y)}$ and its inverse $\Lambda^{-1}$. Formally $d\Lambda(X_t)=\sigma(X_t)^{-1}dX_t=dY_t$, so the candidate solution is $\Lambda^{-1}(\Lambda(X_0)+Y_t-Y_0)$, and the proof shows that $(\Lambda^{-1})'=\sigma$ and that $y\mapsto\sigma(\Lambda^{-1}(y))$ is of locally bounded variation. The second ingredient is a pathwise integral defined through fractional Weyl–Marchaud derivatives and controlled by Gagliardo seminorms; Assumption 2.1 supplies the integrability of $|X_t-y|^{-(\beta+\varepsilon)/\alpha}$ that makes discontinuous functions of $X$ tractable.

What would settle it

Refutation would require a driver satisfying Assumption 2.1 and a coefficient $\sigma$ satisfying Assumption 2.2 for which the candidate process fails (2.1). The simplest check is $Y_t=t^\alpha$ with a two-valued $\sigma$: the pathwise integral is an ordinary Lebesgue integral and the candidate is explicit, so the identity can be evaluated exactly; a mismatch would refute the construction. Separately, a driver with a plateau makes the inverse-distance integral $\sup_y \mathbb{E}\int_0^T |X_t-y|^{-(\beta+\varepsilon)/\alpha}dt$ diverge at the flat level, confirming that the variability condition is essential.

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Extended reading notes

Core claim

Theorem 2.1 states that if the candidate process $Z_t=\Lambda(X_0)+Y_t-Y_0$ satisfies the inverse-distance integrability bound and $\sigma$ satisfies the one-signed locally-bounded-variation condition, then $X_t=\Lambda^{-1}(Z_t)$ is a solution to (2.1). Theorem 2.2 states that every solution satisfying the same integrability condition is unique on $[0,\tau]$, where $\tau=\inf\{t:\sigma(X_t)=0\}$, and that $\tau$ itself is uniquely determined; if $\sigma$ never vanishes, uniqueness holds in that class for all times. Together the theorems give the first general existence-and-uniqueness statement for one-dimensional SDEs driven by Hölder noises with discontinuous diffusion coefficients.

Load-bearing premise

The load-bearing premise is that the relevant process does not linger near any fixed level: expectations of inverse powers of $|X_t-y|$, integrated over time, must be finite uniformly in $y$. If that fails, the pathwise integral of $\sigma(X)$ against $Y$ and the chain rule that produces the solution formula are not justified.

Editorial extensions

If this is right

For every driver satisfying the variability condition—fractional Brownian motion with $H>1/2$, the Rosenblatt process, stationary processes with bounded densities—the SDE has a solution for every one-signed locally bounded-variation $\sigma$ with locally integrable reciprocal.
If $\sigma$ is bounded away from zero, the solution is unique in the class of processes satisfying Assumption 2.1; if $\sigma$ can vanish, uniqueness is guaranteed at least up to the first hitting time of the zero set.
The earlier two-valued discontinuous coefficient and the power-type coefficient $\sigma(x)=|x|^\gamma$ are recovered as special cases, and in the existence part the extra restrictions on $\gamma$ tied to the Hurst parameter disappear.
Examples built from the Cantor function plus a positive constant become uniquely solvable, showing that highly non-smooth but locally bounded-variation coefficients fit the framework.

Reading between the lines

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

Because the construction never uses Markov or martingale structure, the solution is a deterministic functional of the driver path; this suggests a pathwise simulation method: apply $\Lambda^{-1}$ directly to a simulated driver instead of discretizing the SDE.
The inverse-distance condition is likely the real boundary of the method: drivers with flat stretches or paths that concentrate near particular levels fall outside, and non-uniqueness after $\sigma$ hits zero is left open, so the theorem should be read as a statement about the class it explicitly defines.
A natural test of the same mechanism would be equations with drift, $dX_t=b(X_t)\,dt+\sigma(X_t)\,dY_t$; the paper does not treat these, and $\Lambda$ would no longer remove $X$ from the driving term, so the integrability condition would need to be reworked.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is the first general attempt at existence and uniqueness for one-dimensional SDEs dX_t = σ(X_t)dY_t where σ is locally BV and can jump, and Y is Hölder with α > 1/2. The construction via Λ(x) = ∫ 1/σ is classical for smooth σ, and the authors push it through the W^{θ,1} integration spaces from [3], covering fBM with H > 1/2 and the Rosenblatt process. The examples generalizing [6] and [9] are real progress. The writing is honest about what is imported from the companion paper.

But there is a direct counterexample to Theorem 2.1. Take σ(x) = 1 + x² and Y_t = t. Then Λ(x) = arctan x, whose range is bounded. Choose α = 0.75, β = 0.5, ε = 0.1. Assumption 2.1 holds for Z_t = t because ∫_0^T |t−y|^{−0.8}dt is bounded uniformly in y. All other assumptions are satisfied, but the claimed solution is X_t = tan t, which explodes at π/2. The theorem asserts a real-valued solution on the whole interval [0,T]; it fails for T > π/2. The missing condition is that Z_t must stay in the range of Λ, or the statement must be formulated up to the first exit time. This is not a technicality; it is the difference between global existence and a finite-time blow-up.

Secondary issue: the load-bearing approximation result (Prop 4.2) and chain rule (Thm 4.2) are imported from [3] with "we omit the details." That is a serious reliance on a companion paper, though [3] is published and the arguments are plausibly routine. It is not fatal, but it makes verification slow.

If these two issues are corrected—adding the range condition and either proving or precisely citing the deferred machinery—the paper is a solid contribution. As it stands, the main theorem overclaims, and the counterexample should be acknowledged and addressed. I would recommend sending to peer review because the method and scope are valuable and the flaw is repairable. A good referee will catch the same issue immediately; the authors should fix it before publication.

Referee Report

3 major / 4 minor

Summary. The paper studies one-dimensional SDEs of the form dX_t = σ(X_t)dY_t, where Y is a Hölder continuous process of order α > 1/2 (e.g. fractional Brownian motion with H > 1/2 or the Rosenblatt process) and σ is a locally bounded-variation, one-signed function, possibly discontinuous, with 1/σ locally integrable. The main results are Theorem 2.1, which proposes the explicit solution X_t = Λ^{-1}(Λ(X_0)+Y_t−Y_0) with Λ(x)=∫ 1/σ, and Theorem 2.2, which gives uniqueness up to the first hitting time τ = inf{t : σ(X_t)=0} for solutions satisfying an inverse-distance moment condition (Assumption 2.1). The proof combines a pathwise generalized Stieltjes integral (Section 3) with an integration theory for discontinuously evaluated processes (Section 4), several key ingredients of which are imported from the companion paper [3].

Significance. If the range issue identified below is fixed, the paper would provide an elegant and quite general existence/uniqueness theory for one-dimensional SDEs with discontinuous coefficients, going substantially beyond the earlier results of [6] and [9] and covering natural drivers beyond fractional Brownian motion. The explicit solution formula, the absence of fitted parameters, and the treatment of examples such as the Cantor-function coefficient (Example 2.4) are clear strengths. The main obstacles are a missing global range condition in Theorem 2.1 and the fact that several load-bearing results in Section 4 are stated without proofs and are only deferred to [3].

major comments (3)

[Section 2, Theorem 2.1; Section 5.1] The global existence claim is false as stated. Take σ(x)=1+x^2, X_0=0, Y_t=t, and any T>π/2. Then σ satisfies Assumption 2.2, and Z_t=Λ(X_0)+Y_t−Y_0=t satisfies Assumption 2.1 for α∈(1/2,1) (choose β∈(1−α,α) and ε>0 so that (β+ε)/α<1; then sup_y∫_0^T |t−y|^{-(β+ε)/α}dt<∞). The proposed formula gives X_t=Λ^{-1}(t)=tan t, which is undefined at t=π/2, and the ODE dX_t=(1+X_t^2)dt has no global real solution. The theorem must either add an explicit condition that Λ(R)=R, equivalently ∫^∞ 1/σ = ∫_{−∞} 1/σ = ∞, or restate existence only up to the explosion time inf{t : Λ(X_0)+Y_t−Y_0 ∉ range(Λ)}.
[Section 4, Proposition 4.2 and Theorem 4.2] The proofs of Proposition 4.2 and Theorem 4.2 are omitted ('we omit the details'), yet both are load-bearing: Theorem 2.1 applies Theorem 4.2 to Λ^{-1}, and Proposition 4.2 is used in Proposition 5.2 to establish uniqueness. The manuscript states that these follow from [3] after modifications, but the modifications are precisely what is needed to accommodate Assumption 2.1 and the case where f(X_0+) need not exist. Please provide the full proofs or, at minimum, a detailed statement of the modifications; a short deferral to the companion paper is not sufficient for the central chain rule on which the main theorems rest.
[Section 4, Lemma 4.2] The proof of Lemma 4.2 explicitly establishes only boundedness and right-continuity of the displayed functions, with left-continuity relegated to 'the rest of the proof follows as in [3]'. Since the lemma states continuity and is used to justify Proposition 4.2, please include the left-continuity argument or give a precise statement of the corresponding result in [3] that covers this case.

minor comments (4)

[Section 2, Corollaries 2.1 and 2.2] The symbol Z_t is used both for the process Λ(X_0)+Y_t−Y_0 (in Theorem 2.1) and for the solution Λ^{-1}(Λ(X_0)+Y_t−Y_0) (in Corollaries 2.1 and 2.2); please use different letters to avoid confusion.
[Section 5.1, proof of Theorem 2.1] The proof writes sup_z∫_0^T |Z_t−z|^{-β/α}dt, whereas Assumption 2.1 contains an expectation and an exponent −(β+ε)/α; the reduction should be made explicit, especially for random drivers.
[Section 5, Proposition 5.1] The definition of τ_ε reads inf{t : σ(X_s) ≤ ε}; the variable in the condition should be t, not s.
[Corollary 2.2] The phrase 'satisfying 2.1' should read 'satisfying Assumption 2.1'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Lamperti construction and uniqueness proof reduce to a legitimate author-overlapping prior integration theory, not to the paper's own assumptions.

full rationale

The paper's central construction is the standard Lamperti transform: Λ is defined from 1/σ, and the proposed solution is X_t = Λ^{-1}(Λ(X0)+Y_t−Y0). Theorems 2.1 and 2.2 are not obtained by fitting parameters or by renaming an input; they are proved from Assumptions 2.1 and 2.2 using the chain rule Theorem 4.2. The principal non-self-contained ingredient is [3], whose author overlap includes Viitasaari; Theorems 4.1, Proposition 4.2, and Theorem 4.2 are quoted or stated to follow directly from [3]. However, [3] is a published, parameter-free integration theory whose stated assumptions do not include the present SDE result, so the citation is genuine external support rather than a circular self-reference. The present paper extends [3] by applying it to SDEs and by proving uniqueness through the monotonicity of Λ, which are independent contributions. The σ(x)=1+x², Y_t=t objection is a real correctness concern about global existence (since Λ(R)≠R), but it is not a circularity: it shows the theorem may overclaim, not that the claimed solution reduces by definition to the assumptions. Hence no circular step can be exhibited; the derivation chain is not equivalent to its inputs.

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

The paper introduces no free parameters and no invented physical or mathematical entities. It relies on standard fractional calculus and Young integration, on Josephy's composition theorem, and on two explicit domain assumptions: the sufficient variability condition (Assumption 2.1) and the coefficient class (Assumption 2.2). The central machinery from [3] is treated as a black box in several key passages.

assumptions (5)

standard math Zahle's generalized Lebesgue-Stieltjes integral and the associated existence criteria for f in W^{theta,1} and g in W^{1-theta,infinity}.
Used in Section 3 to define the pathwise integral; the paper cites Nualart-Rascanu and Zahle. This is standard machinery for fBm with H > 1/2.
standard math Josephy's theorem characterizing when a composition sigma composed with f is of bounded variation for every BV function sigma.
Used in the proof of Theorem 2.1 to show that y maps to sigma(Lambda^{-1}(y)) is BV; the inverse Lambda^{-1} is increasing, so the preimage condition holds.
domain assumption Assumption 2.1 (sufficient variability): there exists beta in (1-alpha, alpha) and epsilon > 0 with sup_y E integral_0^T |X_t - y|^{-(beta+epsilon)/alpha} dt < infinity.
This is the load-bearing condition that makes integrals of sigma(X_t) exist. It is satisfied by fBm H > 1/2, the Rosenblatt process, and stationary processes with bounded densities, but it is an extra hypothesis on the driver or solution, not a consequence of Holder continuity alone.
domain assumption Assumption 2.2: sigma is locally of bounded variation, one-signed, and 1/sigma is locally integrable.
Defines the coefficient class. It allows jumps and zeros of sigma only on null sets, and it is the problem setting rather than an ad hoc restriction.
domain assumption In Theorem 2.2, uniqueness is asserted only in the class of solutions satisfying Assumption 2.1.
The uniqueness result is explicitly qualified by this class; the paper states this and uses it to prove pathwise agreement before the stopping time tau.

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Cite this review

Pith. "Pith review of Stochastic Differential Equations with Discontinuous Diffusions." pith.science (2026). https://pith.science/paper/G7Z2FGIH

@misc{pith2026190803183,
  author       = {Pith},
  title        = {Pith review of: Stochastic Differential Equations with Discontinuous Diffusions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7Z2FGIH}},
  note         = {Machine review of arXiv:1908.03183}
}

abstract

We study one-dimensional stochastic differential equations of form $dX_t = \sigma(X_t)dY_t$, where $Y$ is a suitable H\"older continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $\sigma$ for which we assume very mild conditions. In particular, we allow $\sigma$ to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

Z. Chen, L. Leskel ¨a, and L. Viitasaari. Pathwise stieltjes integrals of discontinuously evaluated stochastic processes. Stoch. Proc. Appl. , 129(8):2723–2757, 2019

work page 2019

[2] [6]

Garzon, J.A

J. Garzon, J.A. Leon, and S. Torres. Fractional stochastic diﬀ erential equation with dis- continuous diﬀusion. Stochastic Analysis and Applications , 35(6):1113–1123, 2017

work page 2017

[3] [9]

J.A. Leon, D. Nualart, and S. Tindel. Young diﬀerential equations with power type non- linearities. Stoch. Proc. Appl. , 127(9):3042–3067, 2017

work page 2017

[4] [1]

Bass and Z-Q

R. Bass and Z-Q. Chen. One-dimensional stochastic diﬀerential equations with singular and degenerate coeﬃcients. The Indian Journal of Statistics , 67(1):19–45, 2005

work page 2005

[5] [2]

Boufoussi and Y

B. Boufoussi and Y. Ouknine. On a SDE driven by a fractional Bro wnian motion and with monotone drift. Electronic communications in probability , 8(14):122–134, 2003

work page 2003

[6] [4]

H. J. Engelbert and W. Schmidt. On one-dimensional stochastic d iﬀerential equations with generalized drift. Lecture Notes in Control and Inform. Sci. , 69:143–155, 1985

work page 1985

[7] [5]

H. J. Engelbert and W. Schmidt. On solutions of one-dimensional s tochastic diﬀerential equations without drift. Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete., 68(3):287–314, 1985

work page 1985

[8] [7]

M. Josephy. Composing functions of bounded variation. Proceedings of the American Mathematical Society, 83(2):354–356, 1981

work page 1981