Existence and uniqueness for singular stochastic differential equations with piecewise well-behaved coefficients

Beno\^it Nieto; Sara Mazzonetto

arxiv: 2604.21389 · v1 · submitted 2026-04-23 · 🧮 math.PR

Existence and uniqueness for singular stochastic differential equations with piecewise well-behaved coefficients

Sara Mazzonetto , Beno\^it Nieto This is my paper

Pith reviewed 2026-05-09 20:47 UTC · model grok-4.3

classification 🧮 math.PR

keywords generalized stochastic differential equationspathwise uniquenesspasting theoremsingular coefficientsskew sticky diffusionsCox-Ingersoll-Ross processthreshold processesexistence and uniqueness

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The pith

A pasting construction transfers local uniqueness to global pathwise uniqueness for one-dimensional generalized SDEs with singular piecewise coefficients.

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

The paper develops sufficient conditions for pathwise uniqueness of solutions to one-dimensional generalized stochastic differential equations featuring singular coefficients such as distributional drifts and degenerate or discontinuous diffusions. These conditions apply without requiring uniform ellipticity or continuity of the diffusion coefficient, given weak existence and uniqueness in law. The authors introduce a pasting approach that combines solutions of local component equations defined between thresholds into a single global solution while preserving existence and uniqueness properties. This yields the first explicit pasting theorem for pathwise uniqueness in the generalized SDE setting. The results are applied to establish existence and uniqueness for skew sticky threshold Cox-Ingersoll-Ross-type diffusions, including the threshold Chan-Karolyi-Longstaff-Sanders process.

Core claim

We provide sufficient conditions for pathwise uniqueness, under weak existence and uniqueness in law, without assuming uniform ellipticity or continuity of the diffusion coefficient. We also investigate a pasting approach for generalized stochastic differential equations that transfers strong existence and pathwise uniqueness, as well as weak existence and uniqueness in law, from local component equations to a global solution. This provides the first explicit pasting theorem yielding pathwise uniqueness in the setting of generalized stochastic differential equations. As an application, we establish the first existence and uniqueness results for a class of skew sticky threshold Cox-Ingersoll-

What carries the argument

The pasting construction for generalized stochastic differential equations, which glues local solutions on intervals separated by thresholds into a global solution while transferring uniqueness properties.

Load-bearing premise

The local component equations must satisfy the required weak or strong existence and uniqueness properties under the piecewise well-behaved coefficient assumptions.

What would settle it

A concrete counterexample consisting of local equations that each have unique solutions but whose pasted global version admits two distinct solutions would disprove the pasting theorem for pathwise uniqueness.

read the original abstract

We study existence and uniqueness for one-dimensional generalized stochastic differential equations with singular coefficients, including distributional drift and degenerate, possibly discontinuous, diffusion coefficients. Such singularities naturally encode changes in the dynamics at thresholds, including reflecting, skew, or sticky interface behavior. We develop two directions. We provide sufficient conditions for pathwise uniqueness, under weak existence and uniqueness in law, without assuming uniform ellipticity or continuity of the diffusion coefficient. We also investigate a pasting approach for generalized stochastic differential equations that transfers strong existence and pathwise uniqueness, as well as weak existence and uniqueness in law, from local component equations to a global solution. To the best of our knowledge, this provides the first explicit pasting theorem yielding pathwise uniqueness in the setting of generalized stochastic differential equations. As an application, we establish the first existence and uniqueness results for a class of skew sticky threshold Cox-Ingersoll-Ross-type diffusions, including the threshold Chan-Karolyi-Longstaff-Sanders process.

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.

Desk Editor's Note private letter to a colleague

The paper's real contribution is an explicit pasting construction that turns local weak existence plus uniqueness in law into global pathwise uniqueness for 1D generalized SDEs with distributional drifts and discontinuous or degenerate diffusion at thresholds.

read the letter

The main thing to know is that they give a concrete way to glue together solutions on intervals separated by singular points and still keep pathwise uniqueness for the whole process. This works under weaker assumptions than the usual uniform ellipticity or continuous coefficients. They also apply it to get the first existence-uniqueness statements for skew sticky threshold CIR-type processes, including a threshold CKLS variant. That part is new on the face of it and directly useful for models with interface behavior at zero or other levels. The pasting theorem itself is presented as the first explicit one for generalized SDEs, and the conditions they list for transferring strong existence and pathwise uniqueness look workable once the local pieces are in hand. They avoid circularity by relying on standard stochastic analysis tools rather than self-referential definitions. The local-component assumption is the soft spot worth checking: the global result stands only if each piece (standard CIR away from the threshold, modified dynamics right at the interface) already satisfies the required weak or strong properties under the piecewise hypotheses. For the CIR application this needs explicit verification rather than just citation, since the singularity is exactly where standard theorems can fail. If the paper supplies those local checks with the same care as the pasting step, the argument holds; otherwise the application rests on an unproven transfer. This is for people who already work with one-dimensional SDEs that have jumps in coefficients or sticky/reflecting boundaries. A reader who needs to handle threshold models in finance or population dynamics will get a usable tool. It is worth sending to a serious referee because the claims are specific, the method is constructive, and the application is concrete enough to test.

Referee Report

2 major / 1 minor

Summary. The paper studies existence and uniqueness for one-dimensional generalized SDEs with singular coefficients including distributional drifts and degenerate or discontinuous diffusions. It provides sufficient conditions for pathwise uniqueness assuming weak existence and uniqueness in law, without uniform ellipticity or continuity of the diffusion coefficient. It develops a pasting construction that transfers strong existence, pathwise uniqueness, weak existence, and uniqueness in law from local component equations to the global generalized SDE, claimed to be the first explicit such theorem for pathwise uniqueness. As an application, it establishes the first existence and uniqueness results for skew-sticky threshold CIR-type diffusions, including the threshold CKLS process.

Significance. If the pasting theorem and local-to-global transfer hold under the stated conditions, the work would provide a useful framework for analyzing SDEs with threshold singularities that encode reflecting, skew, or sticky behavior. This extends beyond classical assumptions and could apply to models in finance and population dynamics. The application to threshold CIR processes addresses a gap for these processes with interface singularities. The reliance on standard stochastic analysis tools rather than circular constructions is a positive feature.

major comments (2)

The pasting theorem transfers pathwise uniqueness from local component equations to the global generalized SDE, but the manuscript assumes rather than verifies or cites that each local equation (defined on intervals separated by singular thresholds) satisfies the requisite weak/strong existence and uniqueness under the piecewise well-behaved coefficient hypotheses, including cases with distributional drift and degenerate diffusion. This assumption is load-bearing for the global result and the application claims.
In the application to skew-sticky threshold CIR-type diffusions (including threshold CKLS), the local pieces consist of a standard CIR process away from the threshold and modified interface dynamics exactly at the threshold. The paper does not derive or reference a self-contained argument that these local components meet the sufficient conditions for pathwise uniqueness without uniform ellipticity or continuity, which is required for the pasting to yield the claimed global existence and uniqueness.

minor comments (1)

The introduction could include a brief explicit example of a generalized SDE with piecewise coefficients to illustrate the singularities before stating the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify points where the manuscript would benefit from greater explicitness regarding the local components. We will revise the paper to address both major comments by adding clarifications, citations, and brief verifications, without altering the main theorems or the claimed novelty of the global pasting result and the application.

read point-by-point responses

Referee: The pasting theorem transfers pathwise uniqueness from local component equations to the global generalized SDE, but the manuscript assumes rather than verifies or cites that each local equation (defined on intervals separated by singular thresholds) satisfies the requisite weak/strong existence and uniqueness under the piecewise well-behaved coefficient hypotheses, including cases with distributional drift and degenerate diffusion. This assumption is load-bearing for the global result and the application claims.

Authors: We agree that the presentation would be improved by making the local assumptions fully explicit. In the manuscript the local equations are defined on open intervals between thresholds where the coefficients are piecewise well-behaved by construction; they therefore fall under the hypotheses of the pathwise-uniqueness theorem we prove (which requires only weak existence and uniqueness in law, without uniform ellipticity). For distributional-drift cases the local generalized SDEs are covered by the same framework developed in the paper. To remove any ambiguity we will add a short dedicated paragraph (or subsection) that states the local conditions explicitly, recalls the relevant parts of our uniqueness theorem, and supplies citations to classical results for degenerate diffusions where needed. This revision will make the load-bearing step transparent while leaving the global pasting theorem unchanged. revision: yes
Referee: In the application to skew-sticky threshold CIR-type diffusions (including threshold CKLS), the local pieces consist of a standard CIR process away from the threshold and modified interface dynamics exactly at the threshold. The paper does not derive or reference a self-contained argument that these local components meet the sufficient conditions for pathwise uniqueness without uniform ellipticity or continuity, which is required for the pasting to yield the claimed global existence and uniqueness.

Authors: We accept that the application section should contain an explicit link to the local uniqueness conditions. The local CIR (and CKLS) processes on intervals away from the threshold are standard one-dimensional SDEs whose pathwise uniqueness is known from the literature (Feller’s test, Yamada–Watanabe-type arguments adapted to square-root degeneracy, and parameter restrictions ensuring the Feller condition). These local pieces satisfy the “piecewise well-behaved” hypotheses of our general pathwise-uniqueness theorem, which itself does not require uniform ellipticity or continuity. The interface behavior at the threshold is precisely the object of the pasting construction. In the revision we will insert a brief paragraph that (i) recalls the standard uniqueness results for CIR/CKLS, (ii) verifies that they fit our hypotheses, and (iii) notes that the pasting theorem then yields the global result. This supplies the requested self-contained argument while preserving the novelty of the first existence-uniqueness statement for the skew-sticky threshold versions. revision: yes

Circularity Check

0 steps flagged

No circularity; pasting transfers local properties via standard construction without self-definition or load-bearing self-citation.

full rationale

The derivation chain assumes local weak/strong existence and uniqueness for each piecewise component (under the stated coefficient hypotheses) and applies a pasting theorem to obtain global results for the generalized SDE. This is a one-way transfer from independent local inputs to the global object; the local conditions are not derived from the global pasting result, nor are they fitted parameters renamed as predictions. The abstract explicitly frames the pasting theorem as a new explicit result in the generalized setting, supported by external stochastic analysis tools rather than reducing to prior self-citations or ansatzes. No equation or step equates the output to its input by construction. The application to skew-sticky CIR processes similarly verifies local pieces separately. This is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard results from stochastic calculus and probability theory with no new free parameters or invented entities introduced at the abstract level.

axioms (2)

standard math Standard properties of one-dimensional Brownian motion and Ito calculus
Foundation for defining and analyzing the generalized SDEs.
domain assumption Local component equations admit weak or strong solutions under piecewise coefficient conditions
Required for the pasting construction to transfer existence and uniqueness to the global equation.

pith-pipeline@v0.9.0 · 5468 in / 1268 out tokens · 80489 ms · 2026-05-09T20:47:02.247880+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

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Mazzonetto and B

S. Mazzonetto and B. Nieto. Parameters estimation of a threshold Chan-Karolyi- Longstaff-Sanders process from continuous and discrete observations.Scandinavian Journal of Statistics, 52(4):1670–1707, 2025

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Müller-Gronbach, S

T. Müller-Gronbach, S. Sabanis, and L. Yaroslavtseva. Existence, uniqueness and ap- proximation of solutions of SDEs with superlinear coefficients in the presence of discon- tinuities of the drift coefficient.Commun. Math. Sci., 23(2):389–429, 2025

work page 2025
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Nagasawa

M. Nagasawa. Note on pasting of two Markov processes.Séminaire de probabilités de Strasbourg, 10:532–535, 1976

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D. Revuz and M. Yor.Continuous martingales and Brownian motion, volume 293 of Grundlehren Math. Wiss.Berlin: Springer, 3rd ed., 3rd. corrected printing edition, 2005

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M. Salins and K. Spiliopoulos. Markov processes with spatial delay: path space charac- terization, occupation time and properties.Stochastics and Dynamics, 17(06):1750042, 2017

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[26]

Stroock and S

D. Stroock and S. Varadhan.Multidimensional diffusion processes. Springer, 2007

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[27]

Tian and H

Y. Tian and H. Zhang. Skew CIR process, conditional characteristic function, moments and bond pricing.Applied Mathematics and Computation, 329:230–238, 2018

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[28]

G. Trutnau. Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve.Stochastic processes and their appli- cations, 121(8):1845–1863, 2011

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[29]

F. Werner. Concatenation and pasting of right processes.Electronic Journal of Proba- bility, 26(none):1 – 21, 2021. 34

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[30]

Yamada and S

T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations.Journal of Mathematics of Kyoto University, 11(1):155–167, 1971

work page 1971
[31]

Zhang and Y

H. Zhang and Y. Tian. Hitting times for sticky skew CIR process.Stochastics, 96(2): 1052–1071, 2024

work page 2024
[32]

Zhang, L

H. Zhang, L. Tang, F. Wang, and Y. Du. Bond pricing under CIR process with threshold setting.IAENG International Journal of Applied Mathematics, 54(8):1649–1656, 2024. 35

work page 2024

[1] [1]

One-dimensionalstochasticdifferentialequationswithsingular and degenerate coefficients.Sankhy¯ a: The Indian Journal of Statistics, pages 19–45, 2005

R.F.BassandZ.Q.Chen. One-dimensionalstochasticdifferentialequationswithsingular and degenerate coefficients.Sankhy¯ a: The Indian Journal of Statistics, pages 19–45, 2005

work page 2005

[2] [2]

Bass and E

R.F. Bass and E. Pardoux. Uniqueness for diffusions with piecewise constant coefficients. Probability Theory and Related Fields, 76(4):557–572, 1987

work page 1987

[3] [3]

Blei and H.J

S. Blei and H.J. Engelbert. One-dimensional stochastic differential equations with gen- eralized drift.Theory of Probability & Its Applications, 58(3):345–357, 2014

work page 2014

[4] [4]

A. S. Cherny. On the uniqueness in law and the pathwise uniqueness for stochastic differential equations.Theory Probab. Appl., 46(3):406–419, 2001

work page 2001

[5] [5]

Decamps, M

M. Decamps, M. Goovaerts, and W. Schoutens. Self exciting threshold interest rates models.International Journal of Theoretical and Applied Finance, 9(07):1093–1122, 2006

work page 2006

[6] [6]

Dong and H

F. Dong and H. Ying Wong. Longevity bond pricing under the threshold CIR model. Finance Research Letters, 15:195–207, 2015

work page 2015

[7] [7]

Engelbert and G

H.J. Engelbert and G. Peskir. Stochastic differential equations for sticky Brownian motion.Stochastics, 86(6):993–1021, 2014

work page 2014

[8] [8]

Engelbert and W

H.J. Engelbert and W. Schmidt. Strong Markov continuous local martingales and so- lutions of one-dimensional stochastic differential equations (part iii).Mathematische Nachrichten, 151(1):149–197, 1991

work page 1991

[9] [9]

Fukushima, Y

M. Fukushima, Y. Oshima, and M. Takeda.Dirichlet forms and symmetric Markov processes, volume 19. Walter de Gruyter, 2011

work page 2011

[10] [10]

Ikeda, M

N. Ikeda, M. Nagasawa, and S. Watanabe. A construction of Markov processes by piecing out.Proceedings of the Japan Academy, 42(4):370–375, 1966

work page 1966

[11] [11]

Jeanblanc, M

M. Jeanblanc, M. Yor, and M. Chesney.Mathematical methods for financial markets. Springer Science & Business Media, 2009

work page 2009

[12] [12]

Karatzas and S

I. Karatzas and S. Shreve.Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991

work page 1991

[13] [13]

Kopytko and M

B.I. Kopytko and M. Portenko. The problem of pasting together two diffusion processes and classical potentials.Theory of Stochastic Processes, 15(2):126–139, 2009. 33

work page 2009

[14] [14]

Kopytko and N.I

B.I. Kopytko and N.I. Portenko. Analytical methods of pasting together of diffusion processes. InProbability Theory and Mathematical Statistics: Proceedings of the Fourth USSR-Japan Symposium, held at Tbilisi, USSR, August 23–29, 1982, pages 320–326. Springer, 2006

work page 1982

[15] [15]

Lamberton and M

D. Lamberton and M. Zervos. On the optimal stopping of a one-dimensional diffusion. Electronic Journal of Probability, 18(34):1–49, 2013

work page 2013

[16] [16]

J. F. Le Gall. One-dimensional stochastic differential equations involving the local times of the unknown process. In Aubrey Truman and David Williams, editors,Stochastic Analysis and Applications, pages 51–82, Berlin, Heidelberg, 1984. Springer Berlin Hei- delberg

work page 1984

[17] [17]

A. Lejay. The snapping out brownian motion.The Annals of Applied Probability, pages 1727–1742, 2016

work page 2016

[18] [18]

Mazzonetto and B

S. Mazzonetto and B. Nieto. Parameters estimation of a threshold Chan-Karolyi- Longstaff-Sanders process from continuous and discrete observations.Scandinavian Journal of Statistics, 52(4):1670–1707, 2025

work page 2025

[19] [19]

Menoukeu-Pamen, Y

O. Menoukeu-Pamen, Y. Ouknine, and L. Tangpi. Pathwise uniqueness of non-uniformly elliptic sdes with rough coefficients.Journal of Theoretical Probability, 32(4):1892–1908, 2019

work page 1908

[20] [20]

P. Meyer. Renaissance, recollements, mélanges, ralentissement de processus de Markov. InAnnales de l’institut Fourier, volume 25, pages 465–497, 1975

work page 1975

[21] [21]

Mota and M

P. Mota and M. Esquível. On a continuous time stock price model with regime switching, delay, and threshold.Quantitative Finance, 14(8):1479–1488, 2014

work page 2014

[22] [22]

Müller-Gronbach, S

T. Müller-Gronbach, S. Sabanis, and L. Yaroslavtseva. Existence, uniqueness and ap- proximation of solutions of SDEs with superlinear coefficients in the presence of discon- tinuities of the drift coefficient.Commun. Math. Sci., 23(2):389–429, 2025

work page 2025

[23] [23]

Nagasawa

M. Nagasawa. Note on pasting of two Markov processes.Séminaire de probabilités de Strasbourg, 10:532–535, 1976

work page 1976

[24] [24]

Revuz and M

D. Revuz and M. Yor.Continuous martingales and Brownian motion, volume 293 of Grundlehren Math. Wiss.Berlin: Springer, 3rd ed., 3rd. corrected printing edition, 2005

work page 2005

[25] [25]

Salins and K

M. Salins and K. Spiliopoulos. Markov processes with spatial delay: path space charac- terization, occupation time and properties.Stochastics and Dynamics, 17(06):1750042, 2017

work page 2017

[26] [26]

Stroock and S

D. Stroock and S. Varadhan.Multidimensional diffusion processes. Springer, 2007

work page 2007

[27] [27]

Tian and H

Y. Tian and H. Zhang. Skew CIR process, conditional characteristic function, moments and bond pricing.Applied Mathematics and Computation, 329:230–238, 2018

work page 2018

[28] [28]

G. Trutnau. Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve.Stochastic processes and their appli- cations, 121(8):1845–1863, 2011

work page 2011

[29] [29]

F. Werner. Concatenation and pasting of right processes.Electronic Journal of Proba- bility, 26(none):1 – 21, 2021. 34

work page 2021

[30] [30]

Yamada and S

T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations.Journal of Mathematics of Kyoto University, 11(1):155–167, 1971

work page 1971

[31] [31]

Zhang and Y

H. Zhang and Y. Tian. Hitting times for sticky skew CIR process.Stochastics, 96(2): 1052–1071, 2024

work page 2024

[32] [32]

Zhang, L

H. Zhang, L. Tang, F. Wang, and Y. Du. Bond pricing under CIR process with threshold setting.IAENG International Journal of Applied Mathematics, 54(8):1649–1656, 2024. 35

work page 2024