IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v177y2023ics0960077923011207.html
   My bibliography  Save this article

Fractional Brownian motion: Small increments and first exit time from one-sided barrier

Author

Listed:
  • Peng, Qidi
  • Rao, Nan

Abstract

Let {BH(t)}t≥0 be a fractional Brownian motion indexed by non-negative time with initial value BH(0)=0 almost surely and Hurst parameter H∈(0,1). By using a random wavelet series representation of {BH(t)}t≥0, we show that the fractional Brownian increment |BH(t+h)−BH(t)| is almost surely bounded from above by C|h|Hloglog|h|−1 when the time variation |h| is sufficiently small, where the random variable C>0 does not depend on t nor on h; and an upper bound of the p’th moment of C, E[Cp], is provided for each p>0. This result fills some gap in the law of iterated logarithm for fractional Brownian motion, by giving the moments’ control of the almost sure upper bound of fractional Brownian increments. With this enhanced upper bound and some new results on the distribution of the maximum of fractional Brownian motion maxt∈[0,T]BH(t), we obtain a new and refined asymptotic estimate of the upper-tail probability P(τb>T) as T→+∞, where τb is the waiting time for {BH(t)}t≥0 (with H<1/2) to first exit from a positive-valued barrier b.

Suggested Citation

  • Peng, Qidi & Rao, Nan, 2023. "Fractional Brownian motion: Small increments and first exit time from one-sided barrier," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
  • Handle: RePEc:eee:chsofr:v:177:y:2023:i:c:s0960077923011207
    DOI: 10.1016/j.chaos.2023.114218
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923011207
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2023.114218?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Metzler, Adam, 2010. "On the first passage problem for correlated Brownian motion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 277-284, March.
    2. Ran Wang & Yimin Xiao, 2022. "Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2442-2479, December.
    3. Kleyntssens, T. & Nicolay, S., 2022. "From the Brownian motion to a multifractal process using the Lévy–Ciesielski construction," Statistics & Probability Letters, Elsevier, vol. 186(C).
    4. Sixian Jin & Qidi Peng & Henry Schellhorn, 2018. "Estimation of the pointwise Hölder exponent of hidden multifractional Brownian motion using wavelet coefficients," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 113-140, April.
    5. Molchan, G., 2008. "Unilateral small deviations of processes related to the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2085-2097, November.
    6. Aurzada, Frank & Guillotin-Plantard, Nadine & Pène, Françoise, 2018. "Persistence probabilities for stationary increment processes," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1750-1771.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yao Tung Huang & Qingshuo Song & Harry Zheng, 2015. "Weak Convergence of Path-Dependent SDEs in Basket CDS Pricing with Contagion Risk," Papers 1506.00082, arXiv.org, revised May 2016.
    2. Qian, Linyi & Shen, Yang & Wang, Wei & Yang, Zhixin, 2019. "Valuation of risk-based premium of DB pension plan with terminations," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 51-63.
    3. Manfred Marvin Marchione & Enzo Orsingher, 2022. "Hitting Distribution of a Correlated Planar Brownian Motion in a Disk," Mathematics, MDPI, vol. 10(4), pages 1-12, February.
    4. Christian Mönch, 2022. "Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1842-1862, September.
    5. James Brugler & Oliver Linton, 2014. "Circuit Breakers on the London Stock Exchange: Do they improve subsequent market quality?," Cambridge Working Papers in Economics 1453, Faculty of Economics, University of Cambridge.
    6. Peng, Qidi & Zhao, Ran, 2018. "A general class of multifractional processes and stock price informativeness," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 248-267.
    7. Xin Guo & Zhao Ruan & Lingjiong Zhu, 2015. "Dynamics of Order Positions and Related Queues in a Limit Order Book," Papers 1505.04810, arXiv.org, revised Oct 2015.
    8. James Brugler & Oliver Linton, 2014. "Single stock circuit breakers on the London Stock Exchange: do they improve subsequent market quality?," CeMMAP working papers CWP07/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Andrey Itkin & Alexander Lipton, 2017. "Structural default model with mutual obligations," Review of Derivatives Research, Springer, vol. 20(1), pages 15-46, April.
    10. Alexander Lipton & Andrey Gal & Andris Lasis, 2014. "Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1899-1922, November.
    11. Weiping Li & Tim Krehbiel, 2016. "An Improved Approach To Evaluate Default Probabilities And Default Correlations With Consistency," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-29, August.
    12. Aurzada, Frank & Buck, Micha & Kilian, Martin, 2020. "Penalizing fractional Brownian motion for being negative," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6625-6637.
    13. Ahmad, F. & Hambly, B.M. & Ledger, S., 2018. "A stochastic partial differential equation model for the pricing of mortgage-backed securities," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3778-3806.
    14. Wai-Ki Ching & Jia-Wen Gu & Harry Zheng, 2014. "On Correlated Defaults and Incomplete Information," Papers 1409.1393, arXiv.org, revised Jan 2016.
    15. Atkinson, Michael P. & Singham, Dashi I., 2015. "Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 85-92.
    16. Weiping Li, 2016. "Probability of Default and Default Correlations," JRFM, MDPI, vol. 9(3), pages 1-19, July.
    17. Aurzada, Frank & Mukherjee, Sumit, 2023. "Persistence probabilities of weighted sums of stationary Gaussian sequences," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 286-319.
    18. Bras, Pierre & Kohatsu-Higa, Arturo, 2023. "Simulation of reflected Brownian motion on two dimensional wedges," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 349-378.
    19. Tzu-Wei Yang & Lingjiong Zhu, 2015. "A reduced-form model for level-1 limit order books," Papers 1508.07891, arXiv.org, revised Nov 2016.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:177:y:2023:i:c:s0960077923011207. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.