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Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion

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  • Mémin, Jean
  • Mishura, Yulia
  • Valkeila, Esko

Abstract

We study the possibility to control the moments of Wiener integrals of fractional Brownian motion with respect to the Lp- norm of the integrand. It turns out that when the self-similarity index , we can have only an upper inequality, and when we can have only a lower inequality.

Suggested Citation

  • Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    • Handle: RePEc:eee:stapro:v:51:y:2001:i:2:p:197-206
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    References listed on IDEAS

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    1. Novikov, Alexander & Valkeila, Esko, 1999. "On some maximal inequalities for fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 47-54, August.
    2. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
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    Cited by:

    1. Ehsan Azmoodeh & Esko Valkeila, 2013. "Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 16(2), pages 97-112, July.
    2. Lucian Maticiuc & Tianyang Nie, 2015. "Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities," Journal of Theoretical Probability, Springer, vol. 28(1), pages 337-395, March.
    3. Marie, Nicolas, 2020. "Nonparametric estimation of the trend in reflected fractional SDE," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. Fan, Xiliang & Yuan, Chenggui, 2016. "Lyapunov exponents of PDEs driven by fractional noise with Markovian switching," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 39-50.
    5. Nualart, David & Pérez-Abreu, Victor, 2014. "On the eigenvalue process of a matrix fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4266-4282.
    6. Radchenko, Vadym M., 2007. "Besov regularity of stochastic measures," Statistics & Probability Letters, Elsevier, vol. 77(8), pages 822-825, April.
    7. B. L. S. Prakasa Rao, 2021. "Nonparametric Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion with Random Effects," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 554-568, August.
    8. Balan, Raluca M. & Tudor, Ciprian A., 2010. "The stochastic wave equation with fractional noise: A random field approach," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2468-2494, December.
    9. Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    10. Yan, Litan, 2004. "Maximal inequalities for the iterated fractional integrals," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 69-79, August.
    11. Slominski, Leszek & Ziemkiewicz, Bartosz, 2005. "Inequalities for the norms of integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 73(1), pages 79-90, June.
    12. Raluca M. Balan & Ciprian A. Tudor, 2010. "Stochastic Heat Equation with Multiplicative Fractional-Colored Noise," Journal of Theoretical Probability, Springer, vol. 23(3), pages 834-870, September.
    13. Fan, Xiliang & Yu, Ting & Yuan, Chenggui, 2023. "Asymptotic behaviors for distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 383-415.
    14. Slominski, Leszek & Ziemkiewicz, Bartosz, 2009. "On weak approximations of integrals with respect to fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 543-552, February.
    15. Radchenko, Vadym, 2019. "Averaging principle for the heat equation driven by a general stochastic measure," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 224-230.
    16. Dzhaparidze, Kacha & van Zanten, Harry & Zareba, Pawel, 2005. "Representations of fractional Brownian motion using vibrating strings," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1928-1953, December.
    17. Mishura, Yuliya & Shevchenko, Georgiy, 2017. "Small ball properties and representation results," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 20-36.
    18. Rang, Guanglin, 2020. "From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3408-3444.
    19. David Nualart & Youssef Ouknine, 2003. "Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2," Journal of Theoretical Probability, Springer, vol. 16(2), pages 451-470, April.
    20. Fan, Xiliang & Huang, Xing & Suo, Yongqiang & Yuan, Chenggui, 2022. "Distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 23-67.
    21. M. Mishra & B. Prakasa Rao, 2011. "Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 14(2), pages 101-109, May.
    22. Mahmoudi, Fatemeh & Tahmasebi, Mahdieh, 2022. "The convergence of a numerical scheme for additive fractional stochastic delay equations with H>12," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 191(C), pages 219-231.

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