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arxiv: 2404.07013 · v2 · submitted 2024-04-10 · 🧮 math.NA · cs.NA

Numerical approximation of SDEs driven by fractional Brownian motion for all Hin(0,1) using WIS integration

Utku Erdogan , Gabriel J. Lord , Roy B. Schieven This is my paper

Pith reviewed 2026-05-24 02:09 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords SDEfractional Brownian motionWick-Itô-Skorohod integralnumerical approximationstrong convergenceHurst parametertranslation operator
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0 comments X

The pith

A numerical scheme converges for SDEs driven by fractional Brownian motion under WIS integration for every H in (0,1).

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

The paper develops a numerical approximation for quasilinear SDEs with multiplicative fractional Brownian motion, using the Wick-Itô-Skorohod integral so that the stochastic term stays well-defined and centered for every Hurst parameter H between 0 and 1. It extends earlier theory that applied only when H is at least 1/2 by constructing an explicit translation operator that makes the scheme computable for smaller H. Strong convergence is proved, giving an error of order Delta t to the power H in the autonomous case and order Delta t to the power of the minimum of H and a time-Holder exponent otherwise. Experiments suggest the actual rate may reach min(H plus 1/2, 1) for autonomous equations, which would exceed the proven bound. The result makes simulation feasible even when H is small.

Core claim

We prove a strong convergence result that gives, in the autonomous case, an error of O(Δt^H) and in the non-autonomous case O(Δt^min(H,ζ)), where ζ is a time-Hölder continuity parameter, after constructing the translation operator needed to implement the WIS-based method for all H in (0,1).

What carries the argument

The explicitly constructed translation operator that extends the prior WIS numerical method from H ≥ 1/2 to the full interval (0,1).

If this is right

  • The scheme converges strongly at the stated rates once the translation operator is available.
  • Solutions exist and are unique for the WIS-interpreted SDE across all H in (0,1).
  • Practical simulation becomes possible for small H where other integral interpretations are harder to discretize.
  • Observed rates in autonomous cases reach at least min(H + 1/2, 1) in the reported experiments.

Where Pith is reading between the lines

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

  • The gap between proven and observed rates indicates that a sharper analysis could raise the theoretical order without changing the method.
  • The same translation-operator construction might be adapted to other centered integrals or to equations with additive noise.
  • Access to reliable small-H simulations would allow direct numerical checks of models in which rough noise (H near 0) is physically required.

Load-bearing premise

The translation operator remains stable and computable when H is small.

What would settle it

Numerical runs on an autonomous test equation with known exact solution showing that the observed error fails to decrease like O(Δt^H) when H is below 1/2.

Figures

Figures reproduced from arXiv: 2404.07013 by Gabriel J. Lord, Roy B. Schieven, Utku Erdogan.

Figure 1
Figure 1. Figure 1: Sample paths of GBMEM for different values of H using same random numbers for (1.1) with β = 1, a(x) = 4x 1+x2 , x0 = 1, ∆t = 0.001 and (a) α = 0 (MishuraEM) (b) α = 1 (GBMEM). 5.2. Numerical Results. In [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sample paths of the four methods with α = 1, β = 1, a(x) = 4x 1+x2 , x0 = 1, ∆t = 0.001 using same random numbers with (a) H = 0.25, (b) H = 0.75. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The estimated RMSE of the four methods for various values of ∆t for the quasi-linear SDE with parameters in (5.2), α = 1 and T = 1, including a linear fit for the GBMEM method. We have (a) H = 0.25, linear fit slope 0.771 (b) H = 0.75, linear fit slope 1.009. GBMEM (a) with α = 1 and in [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Estimated rates of convergence for β = 1, a(x) = 4x 1+x2 , x0 = 1 and T = 1. The total Monte Carlo sample size is 500, divided in batches of 50 to obtain the given error bars. In (a) α = 0 (MishuraEM) and in (b) α = 1 (GBMEM). (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The estimated rates of convergence for α = 1, β = 2, a(x) = 4x 1+x2 , x0 = 1 and T = 1 (GBMEM). (b) The estimated rates of convergence for α = −1, β = 0.5, a(x) = cos(x), x0 = 10 and T = 1 (GBMEM). In (a) and (b) the total Monte Carlo sample size is 500, divided in batches of 50 to obtain the given error bars [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) The estimated rates of convergence for α = 0, β = 5, a(x) = 25 log(1+x 2 ), x0 = 25 and T = 1 (MishuraEM). The total Monte Carlo sample size is 2500, divided in batches of 250 to obtain the given error bars. (b) The corresponding estimation of the error constant log(CH). 6. Discussion One open question is whether the theoretical rate of convergence can be improved and approach the conjectured estimate … view at source ↗
read the original abstract

We examine the numerical approximation of a quasilinear stochastic differential equation (SDE) with multiplicative fractional Brownian motion. The stochastic integral is interpreted in the Wick-It\^o-Skorohod (WIS) sense that is well defined and centered for all $H\in(0,1)$. We give an introduction to the theory of WIS integration before we examine existence and uniqueness of a solution to the SDE. We then introduce our numerical method which is based on previous theoretical results for $H\geq \frac{1}{2}$. We construct explicitly a translation operator required for the practical implementation of the method and are not aware of any other implementation of a numerical method for the WIS SDE. We then prove a strong convergence result that gives, in the autonomous case, an error of $O(\Delta t^H)$ and in the non-autonomous case $O(\Delta t^{\min(H,\zeta)})$, where $\zeta$ is a time-H\"older continuity parameter. We present some numerical experiments and conjecture that the theoretical results may not be optimal since we observe numerically a rate of $\min(H+\frac{1}{2},1)$ in the autonomous case. This work opens up the possibility to efficiently simulate SDEs for all $H$ values, including small values of $H$ when the stochastic integral is interpreted in the WIS sense.

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 paper develops a numerical method for quasilinear SDEs driven by fractional Brownian motion under the Wick-Itô-Skorohod (WIS) interpretation, valid for all H ∈ (0,1). After recalling existence/uniqueness, it extends prior schemes valid only for H ≥ 1/2 by constructing an explicit translation operator, proves strong convergence of order O(Δt^H) (autonomous) and O(Δt^min(H,ζ)) (non-autonomous), and reports numerical experiments that suggest a possibly sharper observed rate of min(H + 1/2, 1).

Significance. If the translation operator is shown to preserve the necessary moment and regularity properties, the work supplies the first implementable scheme for WIS SDEs at small H, where previous theory was unavailable; the explicit construction and the conjectured sharper rate are potentially useful for rough-path simulations.

major comments (2)
  1. [translation operator construction and convergence proof] The convergence argument for H < 1/2 reduces the problem to the H ≥ 1/2 regime via the explicitly constructed translation operator and then invokes prior theorems; however, no analytic bounds on moments or Hölder regularity of the translated increments, nor any numerical stability verification, are supplied to confirm that the hypotheses of those theorems remain satisfied when H drops below 1/2 (see the paragraph describing the operator and the statement of the convergence theorem).
  2. [convergence theorem] The claimed rates O(Δt^H) and O(Δt^min(H,ζ)) are load-bearing on the operator preserving Wick centering and the required integrability; if those properties fail for small H, the cited prior results no longer apply and the error bounds are unsupported.
minor comments (1)
  1. [numerical experiments] The numerical section reports an observed rate min(H + 1/2, 1) but does not detail the regression procedure or the range of Δt used; adding this information would strengthen the conjecture that the proved rate is not sharp.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our paper. We address each major comment below and propose revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: The convergence argument for H < 1/2 reduces the problem to the H ≥ 1/2 regime via the explicitly constructed translation operator and then invokes prior theorems; however, no analytic bounds on moments or Hölder regularity of the translated increments, nor any numerical stability verification, are supplied to confirm that the hypotheses of those theorems remain satisfied when H drops below 1/2 (see the paragraph describing the operator and the statement of the convergence theorem).

    Authors: We agree that additional details on the properties of the translated process for H < 1/2 would strengthen the argument. The translation operator is constructed explicitly to shift the driving noise such that the WIS integral reduces to a standard Itô integral with respect to a Brownian motion or fBM with H=1/2, preserving the centering and integrability by the properties of the Wick product. However, to rigorously confirm the hypotheses, we will add a new lemma in the revised manuscript that derives the necessary moment bounds and Hölder regularity estimates for the translated increments when H < 1/2, based on the explicit form of the operator. revision: yes

  2. Referee: The claimed rates O(Δt^H) and O(Δt^min(H,ζ)) are load-bearing on the operator preserving Wick centering and the required integrability; if those properties fail for small H, the cited prior results no longer apply and the error bounds are unsupported.

    Authors: The operator is designed to preserve the Wick centering by construction, as the translation is chosen to account for the difference between the WIS and other interpretations. The integrability is inherited from the original fBM process since the operator is a deterministic shift in the appropriate function space. We will revise the convergence theorem statement and proof to explicitly reference these preservation properties and include the supporting estimates as mentioned in the response to the first comment. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence derived from explicit operator and external priors

full rationale

The paper introduces WIS integration theory, proves existence/uniqueness for the SDE, constructs an explicit translation operator to extend prior results valid only for H ≥ 1/2, and then proves the stated strong convergence rates O(Δt^H) or O(Δt^min(H,ζ)). The numerical experiments are presented after the proof and yield an observed rate that the authors conjecture exceeds the proved bound, confirming the derivation chain does not reduce the claimed result to a fitted input, self-definition, or load-bearing self-citation. All load-bearing steps remain independent of the target error bound.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard existence theory for SDEs, the definition of WIS integration, and the construction of the translation operator; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence and uniqueness of solution to the quasilinear SDE under WIS interpretation
    Invoked before the numerical method is introduced.
  • domain assumption Previous theoretical results for H ≥ 1/2 remain valid and can be extended via the translation operator
    Basis for constructing the numerical scheme.

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Solutions of a disease model with fractional white noise

    M. A. Akinlar et al. “Solutions of a disease model with fractional white noise”. In: Chaos Solitons Fractals 137 (2020)

    work page 2020

  2. [2]

    Approximation of Stochastic Volterra Equations with kernels of completely monotone type

    A. Alfonsi and A. Kebaier. “Approximation of Stochastic Volterra Equations with kernels of completely monotone type”. In: Mathematics of Computation 93.346 (2024), pp. 643– 677

    work page 2024

  3. [3]

    An Itˆ o formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter

    C. Bender. “An Itˆ o formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter”. In: Stochastic Process. Appl. 104.1 (2003), pp. 81–106

    work page 2003

  4. [4]

    Pricing by hedging and no-arbitrage beyond semimartingales

    C. Bender, T. Sottinen, and E. Valkeila. “Pricing by hedging and no-arbitrage beyond semimartingales”. In: Finance Stoch. 12.4 (2008), pp. 441–468

    work page 2008

  5. [5]

    On arbitrage-free pricing of weather derivatives based on fractional Brownian motion

    F. Benth. “On arbitrage-free pricing of weather derivatives based on fractional Brownian motion”. In: Applied Mathematical Finance 10 (Feb. 2003), pp. 303–324

    work page 2003

  6. [6]

    A nonlinear parabolic equation with noise. A reduction method

    F. E. Benth and H. K. Gjessing. “A nonlinear parabolic equation with noise. A reduction method”. In: Potential Anal. 12.4 (2000), pp. 385–401

    work page 2000

  7. [7]

    F. E. Benth and J. ˇSaltyt˙ e Benth.Modeling and pricing in financial markets for weather derivatives. Vol. 17. Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013

    work page 2013

  8. [8]

    Minimal variance hedging for fractional Brownian motion

    F. Biagini and B. Øksendal. “Minimal variance hedging for fractional Brownian motion”. In: Methods Appl. Anal. 10.3 (2003), pp. 347–362

    work page 2003

  9. [9]

    Biagini et al

    F. Biagini et al. Stochastic calculus for fractional Brownian motion and applications. Prob- ability and its Applications (New York). Springer-Verlag London, Ltd., London, 2008

    work page 2008

  10. [10]

    Dynamical pricing of weather derivatives

    D. C. Brody, J. Syroka, and M. Zervos. “Dynamical pricing of weather derivatives”. In: Quant. Finance 2.3 (2002), pp. 189–198

    work page 2002

  11. [11]

    Stochastic analysis, rough path analysis and fractional Brownian motions

    L. Coutin and Z. Qian. “Stochastic analysis, rough path analysis and fractional Brownian motions”. In: Probab. Theory Related Fields 122.1 (2002), pp. 108–140

    work page 2002

  12. [12]

    Di Nunno, B

    G. Di Nunno, B. Øksendal, and F. Proske. Malliavin calculus for L´ evy processes with applications to finance. Universitext. Springer-Verlag, Berlin, 2009

    work page 2009

  13. [13]

    Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme

    X.-L. Ding and D. Wang. “Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme”. In: IMA Journal of Numerical Analysis (Apr. 2023)

    work page 2023

  14. [14]

    A general fractional white noise theory and applications to finance

    R. J. Elliott and J. van der Hoek. “A general fractional white noise theory and applications to finance”. In: Math. Finance 13.2 (2003), pp. 301–330. References 30

    work page 2003

  15. [15]

    A new class of exponential integrators for SDEs with multi- plicative noise

    U. Erdogan and G. J. Lord. “A new class of exponential integrators for SDEs with multi- plicative noise”. In: IMA Journal of Numerical Analysis 39.2 (Mar. 2018), pp. 820–846

    work page 2018

  16. [16]

    Forecasting with fractional Brownian motion: a financial perspective

    M. Garcin. “Forecasting with fractional Brownian motion: a financial perspective”. In: Quant. Finance 22.8 (2022), pp. 1495–1512

    work page 2022

  17. [17]

    H. K. Gjessing. A note on the Wick product . Statistical Report No. 23, University of Bergen. 1993

    work page 1993

  18. [18]

    H. K. Gjessing. Wick calculus with applications to anticipating stochastic differential equa- tions. Manuscript, University of Bergen. 1993

    work page 1993

  19. [19]

    Hida et al

    T. Hida et al. White noise . Vol. 253. Mathematics and its Applications. An infinite- dimensional calculus. Kluwer Academic Publishers Group, Dordrecht, 1993

    work page 1993

  20. [20]

    Holden et al

    H. Holden et al. Stochastic partial differential equations. Probability and its Applications. A modeling, white noise functional approach. Birkh¨ auser Boston, Inc., Boston, MA, 1996

    work page 1996

  21. [21]

    Fractional white noise calculus and applications to finance

    Y. Hu and B. Øksendal. “Fractional white noise calculus and applications to finance”. In: Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6.1 (2003), pp. 1–32

    work page 2003

  22. [22]

    Convergence of a numerical scheme associated to stochas- tic differential equations with fractional Brownian motion

    N. Jamshidi and M. Kamrani. “Convergence of a numerical scheme associated to stochas- tic differential equations with fractional Brownian motion”. In: Appl. Numer. Math. 167 (2021), pp. 108–118

    work page 2021

  23. [23]

    J. A. Le´ on, Y. Liu, and S. Tindel. Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates. 2023. arXiv: 2305. 10365

    work page 2023

  24. [24]

    Stochastic analysis of fractional Brownian motions

    S. J. Lin. “Stochastic analysis of fractional Brownian motions”. In: Stochastics Stochastics Rep. 55.1-2 (1995), pp. 121–140

    work page 1995

  25. [25]

    First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case

    Y. Liu and S. Tindel. “First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case”. In: Ann. Appl. Probab. 29.2 (2019), pp. 758–826

    work page 2019

  26. [26]

    G. J. Lord, C. E. Powell, and T. Shardlow. An introduction to computational stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York, 2014

    work page 2014

  27. [27]

    An efficient, three-dimensional, anisotropic, fractional Brownian motion and truncated fractional L´ evy motion simulation algorithm based on successive random additions

    S. Lu, F. Molz, and H.-H. Liu. “An efficient, three-dimensional, anisotropic, fractional Brownian motion and truncated fractional L´ evy motion simulation algorithm based on successive random additions”. In: Computers & Geosciences 29 (2003), pp. 15–25

    work page 2003

  28. [28]

    Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young

    T. Lyons. “Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young”. In: Math. Res. Lett. 1.4 (1994), pp. 451–464

    work page 1994

  29. [29]

    The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion

    Y. Mishura and G. Shevchenko. “The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion”. In: Stochastics 80.5 (2008), pp. 489–511

    work page 2008

  30. [30]

    Quasilinear stochastic differential equations with a fractional-Brownian component

    Y. S. Mishura. “Quasilinear stochastic differential equations with a fractional-Brownian component”. In: Teor. ˘Imovir. Mat. Stat. 68 (2003), pp. 95–106

    work page 2003

  31. [31]

    Y. S. Mishura. Stochastic calculus for fractional Brownian motion and related processes . Vol. 1929. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008

    work page 1929

  32. [32]

    Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

    A. Neuenkirch and I. Nourdin. “Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion”. In: Journal of Theoretical Probability (2007). doi: 10.1007/s10959-007-0083-0

    work page doi:10.1007/s10959-007-0083-0 2007

  33. [33]

    Stochastic integration with respect to fractional Brownian motion and ap- plications

    D. Nualart. “Stochastic integration with respect to fractional Brownian motion and ap- plications”. In: Stochastic models (Mexico City, 2002) . Vol. 336. Contemp. Math. Amer. Math. Soc., Providence, RI, 2003, pp. 3–39

    work page 2002

  34. [34]

    Fractional Brownian motion in finance

    B. Øksendal. “Fractional Brownian motion in finance”. In: Stochastic economic dynamics. Cph. Bus. Sch. Press, Frederiksberg, 2007, pp. 11–56. References 31

    work page 2007

  35. [35]

    Pricing geometric Asian power options under mixed fractional Brownian motion environment

    B. L. S. Prakasa Rao. “Pricing geometric Asian power options under mixed fractional Brownian motion environment”. In: Phys. A 446 (2016), pp. 92–99

    work page 2016

  36. [36]

    A New Approach for Time Series Forecasting: Bayesian Enhanced by Fractional Brownian Motion with Application to Rainfall Series

    C. R. Rivero et al. “A New Approach for Time Series Forecasting: Bayesian Enhanced by Fractional Brownian Motion with Application to Rainfall Series”. In:International Journal of Advanced Computer Science and Applications 7.3 (2016)

    work page 2016

  37. [37]

    Arbitrage with fractional Brownian motion

    L. C. G. Rogers. “Arbitrage with fractional Brownian motion”. In: Math. Finance 7.1 (1997), pp. 95–105

    work page 1997

  38. [38]

    Measuring anti-correlations in the nordic electricity spot market by wavelets

    I. Simonsen. “Measuring anti-correlations in the nordic electricity spot market by wavelets”. In: Physica A: Statistical Mechanics and its Applications 322 (2003), pp. 597–606

    work page 2003

  39. [39]

    Thangavelu

    S. Thangavelu. Lectures on Hermite and Laguerre expansions. Vol. 42. Mathematical Notes. With a preface by Robert S. Strichartz. Princeton University Press, Princeton, NJ, 1993

    work page 1993

  40. [40]

    Stock market prices and long-range dependence

    W. Willinger, M. S. Taqqu, and V. Teverovsky. “Stock market prices and long-range dependence”. In: Finance and Stochastics 3 (1999), pp. 1–13

    work page 1999

  41. [41]

    Pricing currency options in a fractional Brownian motion with jumps

    W.-L. Xiao et al. “Pricing currency options in a fractional Brownian motion with jumps”. In: Economic Modelling 27 (2010), pp. 935–942

    work page 2010

  42. [42]

    Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

    S.-Q. Zhang and C. Yuan. “Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation”. In: Proc. Roy. Soc. Edinburgh Sect. A 151.4 (2021), pp. 1278–1304. Department Of Mathematics, Eskisehir Technical University, Eskisehir, Turkiye Email address : utkuerdogan@eskisehir.edu.tr ...

    work page 2021