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Unilateral small deviations of processes related to the fractional Brownian motion

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  • Molchan, G.

Abstract

Let x(s), s[set membership, variant]Rd be a Gaussian self-similar random process of index H. We consider the problem of log-asymptotics for the probability pT that x(s), x(0)=0 does not exceed a fixed level in a star-shaped expanding domain T[dot operator][Delta] as T-->[infinity]. We solve the problem of the existence of the limit, [theta]:=lim(-logpT)/(logT)D, T-->[infinity], for the fractional Brownian sheet x(s), s[set membership, variant][0,T]2 when D=2, and we estimate [theta] for the integrated fractional Brownian motion when D=1.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:11:p:2085-2097
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    References listed on IDEAS

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    1. Csáki, Endre & Khoshnevisan, Davar & Shi, Zhan, 2000. "Boundary crossings and the distribution function of the maximum of Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 1-18, November.
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    Cited by:

    1. 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).

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