Conditional stochastic differential equations driven by fractional Brownian motion

Bernt {\O}ksendal; Jasmina {\DJ}or{\dj}evi\'c

arxiv: 2306.08324 · v4 · pith:NYPRDC4Rnew · submitted 2023-06-14 · 🧮 math.PR · math.FA

Conditional stochastic differential equations driven by fractional Brownian motion

Jasmina {\DJ}or{\dj}evi\'c , Bernt {\O}ksendal This is my paper

Pith reviewed 2026-05-25 08:36 UTC · model grok-4.3

classification 🧮 math.PR math.FA

keywords fractional Brownian motionstochastic differential equationsexistence and uniquenessWIS integralconditional SDEHurst parameterwhite noise analysisL2 solutions

0 comments

The pith

Conditional WIS-stochastic differential equations driven by fractional Brownian motion with H > 1/2 have unique solutions in L²(P) under Lipschitz conditions on the coefficients.

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

The paper proves existence and uniqueness for solutions to conditional WIS-SDEs driven by fractional Brownian motion when the Hurst index exceeds 1/2. It first recalls the theory of fractional white noise and establishes a basic L² bound on WIS-integrals. These ingredients are combined to close a fixed-point argument showing that Lipschitz coefficients guarantee a unique L²(P) solution. A reader would care because fractional Brownian motion models systems with long-range dependence, and the conditional setting incorporates additional information constraints common in applications.

Core claim

We prove the existence and uniqueness of a solution in L²(P) of a conditional WIS-stochastic differential equation driven by a fractional Brownian motion with H>1/2 under Lipschitz conditions on its coefficients.

What carries the argument

The WIS-integral (Wick-Itô-Skorohod integral) driven by fractional white noise, together with the L²-estimate that controls its norm and enables the contraction mapping.

If this is right

Solutions can be obtained as the L²-limit of Picard iterates.
The result applies directly to conditional problems that incorporate partial observations.
The L²-estimate for WIS-integrals extends to other linear functionals of the fractional noise.
Existence holds pathwise in the L² sense rather than almost-surely for each path.

Where Pith is reading between the lines

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

The same L² framework could be tested on equations with jumps or regime switches that preserve the fractional driver.
Numerical discretizations that respect the WIS-integral definition might inherit the uniqueness from the continuous theory.
Comparison of this conditional WIS solution with the corresponding Stratonovich or pathwise integral versions would quantify the effect of the interpretation choice.

Load-bearing premise

The coefficients satisfy Lipschitz conditions.

What would settle it

Construct a specific non-Lipschitz coefficient pair for which the Picard iteration fails to converge in L²(P) or produces distinct solutions when H>1/2.

read the original abstract

The aim of this paper is to analyse a WIS-stochastic differential equation driven by fractional Brownian motion with $H>\tfrac{1}{2}$. For this, we summarise the theory of fractional white noise and prove a fundamental $L^2$-estimate for WIS-integrals. We apply this to prove the existence and uniqueness of a solution in $L^2(P)$ of a conditional WIS-stochastic differential equation driven by a fractional Brownian motion with $H>\tfrac{1}{2}$ under Lipschitz conditions on its coefficients.

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 proves existence and uniqueness in L2 for conditional WIS-SDEs driven by fBM (H>1/2) under Lipschitz conditions by first deriving an L2 estimate for the integrals.

read the letter

The central result is an existence-uniqueness theorem for solutions in L2(P) to conditional WIS-stochastic differential equations driven by fractional Brownian motion with H greater than 1/2. The authors first summarize fractional white noise theory, establish a basic L2 estimate for WIS-integrals, and then apply standard arguments under the usual Lipschitz assumption on the coefficients to close the proof. This matches the structure they outline in the abstract and appears to be a direct extension of prior white-noise work to the conditional fractional setting. The L2 estimate is the new technical step that enables the application, and the hypotheses line up with what is common in this literature. The argument does not show obvious circularity or reliance on fitted quantities. The main limitation is that the abstract gives no explicit form or proof sketch for the L2 estimate, so it is not possible to verify from the given material whether the Lipschitz condition alone suffices or whether extra regularity on the driving noise or coefficients is required. The paper is written for specialists already working with white-noise calculus and fractional processes. A reader in that niche can extract a usable existence result and the supporting estimate. Outside that group the contribution is narrow. The work shows clear engagement with the relevant prior theory and states its claims without internal contradiction, so it is worth sending to a referee even though the advance is incremental rather than foundational.

Referee Report

0 major / 2 minor

Summary. The paper summarizes the theory of fractional white noise, proves a fundamental L²-estimate for WIS-integrals, and applies this to prove the existence and uniqueness in L²(P) of solutions to conditional WIS-stochastic differential equations driven by fractional Brownian motion with H > 1/2 under Lipschitz conditions on the coefficients.

Significance. If the L² estimate holds under the stated hypotheses, the work supplies a useful technical tool for the white-noise approach to fractional SDEs and extends it to the conditional setting; the estimate itself may have independent value for further estimates or approximations in this framework.

minor comments (2)

The abstract states that an L² estimate is proved and then applied but does not record the estimate itself; adding a one-sentence formulation of the estimate (with its precise hypotheses) would make the logical structure immediately verifiable from the abstract alone.
Notation for the conditional WIS-integral and the underlying probability space should be introduced once in a dedicated preliminary subsection rather than scattered across the summary of prior theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work, the assessment of its significance as a technical tool in the white-noise approach to fractional SDEs, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first summarizes fractional white-noise theory (standard background) and proves a new L²-estimate for WIS-integrals under H>1/2. It then applies this estimate via the standard Picard iteration / contraction mapping argument under Lipschitz coefficients to obtain existence/uniqueness in L²(P). No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain; the central theorem rests on an independently derived estimate plus classical SDE techniques. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the Lipschitz condition is the only explicit modeling assumption visible.

axioms (1)

domain assumption Coefficients satisfy Lipschitz conditions
Invoked to obtain uniqueness after the L² estimate; standard in SDE theory but load-bearing here.

pith-pipeline@v0.9.0 · 5617 in / 1159 out tokens · 28061 ms · 2026-05-25T08:36:38.702575+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove the existence and uniqueness of a solution in L²(P) of a conditional WIS-stochastic differential equation driven by a fractional Brownian motion with H>1/2 under Lipschitz conditions on its coefficients.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1 ... E[X²(t)] ≤ K(H) E[∫ ϕ²(s) dt] t^{2H-1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 1 internal anchor

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Bainov, D. & Simeonov, P. (1992): Integral Inequalities and Appli- cations, Kluwer Academic Publishers, Dordrecht, Netherla nds. 21

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Biagini, F., Øksendal, B. (2008): Forward integrals and an ˆIto for- mula for fractional Brownian motion, Inﬁnite Dimensional A nalysis, Quantum Probabilityand Related Topics Vol. 11, No. 2, 1–21

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& Øksendal, B

Biagini, F. & Øksendal, B. (2008): Forward integrals and an Itô for- mula for fractional Brownian motion. Inﬁnite Dim. Analysis , Quan- tum Probab. and Related Topics 11, 157-177

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Boudaoui, A., Caraballo, T. & Ouahab, A. (2017): Stochas tic diﬀerential equations with non-instantaneous impulses dr iven by a fractional Brownian motion, Discrete and Continuous Dyna m- ical Systems - Series B, Volume 22, Issue 7: 2521-2541. Doi: 10.3934/dcdsb.2017084

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Decreussefond, L. & Ustunel, A.S. (1999): Stochastic A nalysis of the FractionalBrownian Motion, Potential Analysis 10: 177-21 4

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Eliazar, I. & Kachman, T. ( 2022): Anoimalous diﬀusion: fractional Brownian motion vs fractional Ito motion. J. Phys. A: Math. T heor. 55 115002 22

work page 2022
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Elliot, R. & var det Hoek, J. (2003): A general fractiona l white noise theory and applications to ﬁnance. Mathematical Fina nce 13 (2), 301-330

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& Nualart, D

Guerrera, J. & Nualart, D. (2008): Stochastic diﬀerent ial equations driven by fractional Brownian motion and standard Brownian mo- tion, Stochastic Analysis and Applications Volume 26, Issu e 5, 1053– 1075

work page 2008
[14]

It\^o type stochastic differential equations driven by fractional Brownian motions of Hurst parameter $H>1/2$

Hu, Y. (2016): Ito type stochastic diﬀerential equatio ns driven by fractional Brownian motion of Hurst parameter H > 1/2. https://doi.org/10.48550/arXiv.1610.01137

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1610.01137 2016
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Bernoulli 15(3), 840-870

(2009): Stochastic diﬀerential equations driven by fr actional Brow- nian motion. Bernoulli 15(3), 840-870

work page 2009
[16]

& Ebadi, M.J

Jafari, H., Malinowski, M.T. & Ebadi, M.J. (2021): Fuzz y stochastic diﬀerential equations driven by fractional Brownian motio n. Adv. Diﬀ. Eq. 2021, 16. https://doi.org/10.1186/s13662-020-0 3181-z

work page doi:10.1186/s13662-020-0 2021
[17]

(2009): Stochastic diﬀerential equat ions driven by fractional Brownian motions, Bernoulli Vol

Jien, Y., & Ma, J. (2009): Stochastic diﬀerential equat ions driven by fractional Brownian motions, Bernoulli Vol. 15, No. 3, pp. 8 46–870

work page 2009
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(1968): Fractional Browni an motions, fractional noises and applications

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(2011): Stochastic Diﬀerential Equations and A pplications, Woodhea Publishing Limited

Mao, X. (2011): Stochastic Diﬀerential Equations and A pplications, Woodhea Publishing Limited

work page 2011
[21]

& Valkeila, E

Mémina, J., Mishura, Y. & Valkeila, E. (2001): Inequalities for the moments of Wiener integrals with respect to a fractional Bro wnian motion, Statistics & Probability Letters 51 (2001) 197–206

work page 2001
[22]

(2008): Stochastic Calculus for Fractiona l Brownian Mo- tion and Related Processes

Mishura, Y. (2008): Stochastic Calculus for Fractiona l Brownian Mo- tion and Related Processes. Springer Lecture Notes in Mathe matics no. 1929. 23

work page 2008
[23]

(2023): Asymptotic inferenc e for stochas- tic diﬀerential equations driven by fractional Brownian mo tion

Nakajima, S., Shimizu, Y. (2023): Asymptotic inferenc e for stochas- tic diﬀerential equations driven by fractional Brownian mo tion. Jpn J Stat Data Sci 6, 431-455. https://doi.org/10.1007/s42081 -022-00181- z

work page doi:10.1007/s42081 2023
[24]

& Rascanu, A

Nualart, D. & Rascanu, A. (2002): Stochastic diﬀerential eq uations driven by fractional Brownian motion. Collect. Math. 53 (1) , 55-81

work page 2002
[25]

(2013): Stochastic Diﬀerential Equation s

Øksendal, B. (2013): Stochastic Diﬀerential Equation s. 6th edition

work page 2013
[26]

& Sulem, A

Øksendal, B. & Sulem, A. (2019): Applied Stochastic Con trol of Jump Diﬀusions. Third Edition. Springer

work page 2019
[27]

and Taqqu, M.S

Samorodnitsky, G. and Taqqu, M.S. (1994) Stable Non-Ga ussian Random Processes. Stochastic Models with Inﬁnite Variance . Stochastic Modeling. Chapman & Hall, New York

work page 1994
[28]

(2022): Stochastic delayed fraction al-order dif- ferential equations driven by fractional Brownian motion, MALAYA JOURNAL OF MATEMATIK Malaya J

Sayed Ahmed, A.M. (2022): Stochastic delayed fraction al-order dif- ferential equations driven by fractional Brownian motion, MALAYA JOURNAL OF MATEMATIK Malaya J. Mat.10(03), 187-197. http://doi.org/10.26637/mjm1003/00

work page doi:10.26637/mjm1003/00 2022
[29]

Viens, F

Tindel, S., Tudor, C. Viens, F. (2003): Stochastic evol ution equa- tions with fractional Brownian motion. Probab. Theory Rela t. Fields 127, 186-204 . https://doi.org/10.1007/s00440-003-0282 -2

work page doi:10.1007/s00440-003-0282 2003
[30]

(1996): A general existence and uniqueness the orem for Wick-SDEs in (Y )n 1

Våge, G. (1996): A general existence and uniqueness the orem for Wick-SDEs in (Y )n 1 . Stoch. Stoch. Reports 58,259-284

work page 1996
[31]

(2025): A veraging p rinci- ple for McKean-Vlasov SDEs driven by FBMs

Zhang, T., Xu, Y., Feng, L., & Pei, B. (2025): A veraging p rinci- ple for McKean-Vlasov SDEs driven by FBMs. Qualitative Theo ry of Dynamical Systems, issue 2. https://doi.org/10.1007/s 12346-024- 01099-5 24

work page doi:10.1007/s 2025

[1] [1]

& Simeonov, P

Bainov, D. & Simeonov, P. (1992): Integral Inequalities and Appli- cations, Kluwer Academic Publishers, Dordrecht, Netherla nds. 21

work page 1992

[2] [2]

(2008): Forward integrals and an ˆIto for- mula for fractional Brownian motion, Inﬁnite Dimensional A nalysis, Quantum Probabilityand Related Topics Vol

Biagini, F., Øksendal, B. (2008): Forward integrals and an ˆIto for- mula for fractional Brownian motion, Inﬁnite Dimensional A nalysis, Quantum Probabilityand Related Topics Vol. 11, No. 2, 1–21

work page 2008

[3] [3]

(2008): Sto chas- tic Calculus for Fractional Brownian Motion and Applicatio ns

Biagini, F., Hu, Y., Øksendal, B., & Zhang, T. (2008): Sto chas- tic Calculus for Fractional Brownian Motion and Applicatio ns. Springer

work page 2008

[4] [4]

(2003): An Ito formula for generalised functi onals of a fractional Brownian motion with arbitrary Hurst parame ter

Bender, C. (2003): An Ito formula for generalised functi onals of a fractional Brownian motion with arbitrary Hurst parame ter. Stochastic Proc. App.. 104 (1), 81-106

work page 2003

[5] [5]

(2003): An S-transform approach to integrati on with respect to a fractional Brownian motion, Bernoulli 9(6), 95 5–983

Bender, C. (2003): An S-transform approach to integrati on with respect to a fractional Brownian motion, Bernoulli 9(6), 95 5–983

work page 2003

[6] [6]

& Øksendal, B

Biagini, F. & Øksendal, B. (2008): Forward integrals and an Itô for- mula for fractional Brownian motion. Inﬁnite Dim. Analysis , Quan- tum Probab. and Related Topics 11, 157-177

work page 2008

[7] [7]

& Ouahab, A

Boudaoui, A., Caraballo, T. & Ouahab, A. (2017): Stochas tic diﬀerential equations with non-instantaneous impulses dr iven by a fractional Brownian motion, Discrete and Continuous Dyna m- ical Systems - Series B, Volume 22, Issue 7: 2521-2541. Doi: 10.3934/dcdsb.2017084

work page doi:10.3934/dcdsb.2017084 2017

[8] [8]

& Coutin, L

Carmona, P. & Coutin, L. (1998): Fractional Brownian mot ion and the Markov property, Elect. Comm. in Probab. 3, 95–107

work page 1998

[9] [9]

, Zhibo, Z

Zhao, C. , Zhibo, Z. & Qinghui, D. (2021): Optimal control of stochastic system with fractional Brownian motion, Mathem atical Biosciences and Engineering, vol. 18, no. 5, Sept.-Oct. , pp . 5625- 5634

work page 2021

[10] [10]

& Ustunel, A.S

Decreussefond, L. & Ustunel, A.S. (1999): Stochastic A nalysis of the FractionalBrownian Motion, Potential Analysis 10: 177-21 4

work page 1999

[11] [11]

& Kachman, T

Eliazar, I. & Kachman, T. ( 2022): Anoimalous diﬀusion: fractional Brownian motion vs fractional Ito motion. J. Phys. A: Math. T heor. 55 115002 22

work page 2022

[12] [12]

& var det Hoek, J

Elliot, R. & var det Hoek, J. (2003): A general fractiona l white noise theory and applications to ﬁnance. Mathematical Fina nce 13 (2), 301-330

work page 2003

[13] [13]

& Nualart, D

Guerrera, J. & Nualart, D. (2008): Stochastic diﬀerent ial equations driven by fractional Brownian motion and standard Brownian mo- tion, Stochastic Analysis and Applications Volume 26, Issu e 5, 1053– 1075

work page 2008

[14] [14]

It\^o type stochastic differential equations driven by fractional Brownian motions of Hurst parameter $H>1/2$

Hu, Y. (2016): Ito type stochastic diﬀerential equatio ns driven by fractional Brownian motion of Hurst parameter H > 1/2. https://doi.org/10.48550/arXiv.1610.01137

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1610.01137 2016

[15] [15]

Bernoulli 15(3), 840-870

(2009): Stochastic diﬀerential equations driven by fr actional Brow- nian motion. Bernoulli 15(3), 840-870

work page 2009

[16] [16]

& Ebadi, M.J

Jafari, H., Malinowski, M.T. & Ebadi, M.J. (2021): Fuzz y stochastic diﬀerential equations driven by fractional Brownian motio n. Adv. Diﬀ. Eq. 2021, 16. https://doi.org/10.1186/s13662-020-0 3181-z

work page doi:10.1186/s13662-020-0 2021

[17] [17]

(2009): Stochastic diﬀerential equat ions driven by fractional Brownian motions, Bernoulli Vol

Jien, Y., & Ma, J. (2009): Stochastic diﬀerential equat ions driven by fractional Brownian motions, Bernoulli Vol. 15, No. 3, pp. 8 46–870

work page 2009

[18] [18]

(1993): Fractional Brownian ﬁelds as inte grals of white noise

Linstrøm, T. (1993): Fractional Brownian ﬁelds as inte grals of white noise. Bull. London Math. Soc. 35, 83-88

work page 1993

[19] [19]

(1968): Fractional Browni an motions, fractional noises and applications

Mandelbrot, B., Van Ness, J. (1968): Fractional Browni an motions, fractional noises and applications. SIAM Review, 10(4), 42 2-437

work page 1968

[20] [20]

(2011): Stochastic Diﬀerential Equations and A pplications, Woodhea Publishing Limited

Mao, X. (2011): Stochastic Diﬀerential Equations and A pplications, Woodhea Publishing Limited

work page 2011

[21] [21]

& Valkeila, E

Mémina, J., Mishura, Y. & Valkeila, E. (2001): Inequalities for the moments of Wiener integrals with respect to a fractional Bro wnian motion, Statistics & Probability Letters 51 (2001) 197–206

work page 2001

[22] [22]

(2008): Stochastic Calculus for Fractiona l Brownian Mo- tion and Related Processes

Mishura, Y. (2008): Stochastic Calculus for Fractiona l Brownian Mo- tion and Related Processes. Springer Lecture Notes in Mathe matics no. 1929. 23

work page 2008

[23] [23]

(2023): Asymptotic inferenc e for stochas- tic diﬀerential equations driven by fractional Brownian mo tion

Nakajima, S., Shimizu, Y. (2023): Asymptotic inferenc e for stochas- tic diﬀerential equations driven by fractional Brownian mo tion. Jpn J Stat Data Sci 6, 431-455. https://doi.org/10.1007/s42081 -022-00181- z

work page doi:10.1007/s42081 2023

[24] [24]

& Rascanu, A

Nualart, D. & Rascanu, A. (2002): Stochastic diﬀerential eq uations driven by fractional Brownian motion. Collect. Math. 53 (1) , 55-81

work page 2002

[25] [25]

(2013): Stochastic Diﬀerential Equation s

Øksendal, B. (2013): Stochastic Diﬀerential Equation s. 6th edition

work page 2013

[26] [26]

& Sulem, A

Øksendal, B. & Sulem, A. (2019): Applied Stochastic Con trol of Jump Diﬀusions. Third Edition. Springer

work page 2019

[27] [27]

and Taqqu, M.S

Samorodnitsky, G. and Taqqu, M.S. (1994) Stable Non-Ga ussian Random Processes. Stochastic Models with Inﬁnite Variance . Stochastic Modeling. Chapman & Hall, New York

work page 1994

[28] [28]

(2022): Stochastic delayed fraction al-order dif- ferential equations driven by fractional Brownian motion, MALAYA JOURNAL OF MATEMATIK Malaya J

Sayed Ahmed, A.M. (2022): Stochastic delayed fraction al-order dif- ferential equations driven by fractional Brownian motion, MALAYA JOURNAL OF MATEMATIK Malaya J. Mat.10(03), 187-197. http://doi.org/10.26637/mjm1003/00

work page doi:10.26637/mjm1003/00 2022

[29] [29]

Viens, F

Tindel, S., Tudor, C. Viens, F. (2003): Stochastic evol ution equa- tions with fractional Brownian motion. Probab. Theory Rela t. Fields 127, 186-204 . https://doi.org/10.1007/s00440-003-0282 -2

work page doi:10.1007/s00440-003-0282 2003

[30] [30]

(1996): A general existence and uniqueness the orem for Wick-SDEs in (Y )n 1

Våge, G. (1996): A general existence and uniqueness the orem for Wick-SDEs in (Y )n 1 . Stoch. Stoch. Reports 58,259-284

work page 1996

[31] [31]

(2025): A veraging p rinci- ple for McKean-Vlasov SDEs driven by FBMs

Zhang, T., Xu, Y., Feng, L., & Pei, B. (2025): A veraging p rinci- ple for McKean-Vlasov SDEs driven by FBMs. Qualitative Theo ry of Dynamical Systems, issue 2. https://doi.org/10.1007/s 12346-024- 01099-5 24

work page doi:10.1007/s 2025