Abstract.
Let B be a fractional Brownian motion with Hurst index H∈(0,1). Denote by
the positive, real zeros of the Bessel function J
−H
of the first kind of order −H, and let
be the positive zeros of J
1−H
. In this paper we prove the series representation
where X
1
,X
2
,... and Y
1
,Y
2
,... are independent, Gaussian random variables with mean zero and
and the constant c
H
2 is defined by c
H
2=π−1Γ(1+2H) sin πH. We show that with probability 1, both random series converge absolutely and uniformly in t∈[0,1], and we investigate the rate of convergence.
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Mathematics Subject Classification (2000): 60G15, 60G18, 33C10
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Dzhaparidze, K., Zanten, H. A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130, 39–55 (2004). https://doi.org/10.1007/s00440-003-0310-2
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DOI: https://doi.org/10.1007/s00440-003-0310-2
Keywords
- Fractional Brownian motion
- Series expansion
- Bessel function
- Hankel transform
- Fractional calculus