Forward and symmetric Wick-Itô integrals with respect to fractional Brownian motion

Abstract

Let B^H=\{{{\displaystyle B}}_t^H,t\ge \mathrm{0}\} be a fractional Brownian motion with Hurst index H\in (\mathrm{0},\mathrm{1}). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to B^H as follows:

{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{-}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_s^H)\mathrm{d}s,

{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{°}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_{(s-\epsilon )\vee \mathrm{0}}^H)\mathrm{d}s,

in probability, where ◊ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of B^H for all H\in (\mathrm{0},\mathrm{1}).

Keywords

Fractional Brownian motion (fBm) / forward integral / Malliavin calculus / Wick product

References

[1] Alós E, León J,Nualart D. Stratonovich stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2. Taiwanesse J Math, 2001, 5: 609–632

[2] Alós E, Mazet O, Nualart D. Stochastic calculus with respect to Gaussian processes. Ann Probab, 2001, 29: 766–801

[3] Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic Calculus for Fractional Brownian Motion and Applications. Berlin and New York: Springer, 2008

[4] Cheridito P,Nualart D. Stochastic integral of divergence type with respect to the fractional Brownian motion with Hurst parameter 1/2. Ann Inst Henri Poincaré Probab Stat, 2005, 41: 1049–1081

[5] Gradinaru M, Nourdin I, Russo F, Vallois P.m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index. Ann Inst Henri Poincaró Probab Stat, 2005, 41: 781–806

[6] Gradinaru M, Russo F,Vallois P.Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H≥1/4. Ann Probab, 2003, 31: 1772–1820

[7] Hu Y Z. Integral Transformations and Anticipative Calculus for Fractional Brownian Motions. Mem Amer Math Soc, Vol 175, No 825. Providence: Amer Math Soc, 2005

[8] Mishura Y S. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Berlin: Springer, 2008

[9] Nourdin I. Selected Aspects of Fractional Brownian Motion.Berlin:Springer-Verlag, 2012

[10] Nualart D.Stochastic integration with respect to fractional Brownian motion and applications. Contemp Math, 2003, 336: 3–39

[11] Nualart D. The Malliavin Calculus and Related Topics.Berlin: Springer-Verlag, 2006

[12] Nualart D,Taqqu M S. Wick-Itô formula for Gaussian processes. Stoch Anal Appl, 2006, 24: 599–614

[13] Pipiras V, Taqqu M. Integration questions related to the fractional Brownian motion. Probab Theory Related Fields, 2001, 118: 251–281

[14] Russo F, Vallois P. Forward, backward and symmetric stochastic integration. Probab Theory Related Fields, 1993, 97: 403–421

[15] Russo F,Vallois P. The generalized covariation process and Itô formula. Stochastic Process Appl, 1995, 59: 81–104

[16] Russo F, Vallois P. Stochastic calculus with respect to continuous finite quadratic variation processes. Stochast Stochast Rep, 2000, 70: 1–40

[17] Sun X, Yan L, Yu X.An integral functional driven by fractional Brownian motion. Stochastic Process Appl, 2019, 129: 2249–2285

[18] Yan L. The fractional derivative for fractional Brownian local time. Math Z, 2016, 283: 437–468

[19] Yan L, Liu J, Chen C. The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2. Infin Dimens Anal Quantum Probab Relat Top, 2014, 17(4): 1{32

[20] Yan L, Liu J, Yang X. Integration with respect to fraction local time with Hurst index 1/2<H<1. Potential Anal, 2009, 30: 115–138

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