2002

# A series expansion of fractional Brownian motion with Hurst index exceeding 1/2

## Publication

### Publication

Let $B$ be a fractional Brownian motion with Hurst index $H ge 1/2$. Denote by $x_1 < x_2 < cdots$ the positive, real zeros of the Bessel function $J_{-H$ of the first kind of order $-H$, and by $y_1 < y_2 < cdots$ the positive zeros of $J_{1-H$. We prove the series representation begin{equation* B_t = sum_{n=1^infty frac{sin x_n t{x_n, X_n + sum_{n=1^infty frac{1-cos y_n t{y_n, Y_n, end{equation* where $X_1, X_2, ldots$ and $Y_1, Y_2, ldots$ are independent, Gaussian random variables with mean zero and $Var X_n = 2c_H^2x_n^{-2HJ^{-2_{1-H(x_n)$, $Var Y_n = 2c_H^2y_n^{-2HJ^{-2_{-H(y_n)$, where the constant $c_H^2$ is defined by $c_H^2 = pi^{-12HGamma(2H)sin pi H$. With probability $1$, the random series converges absolutely and uniformly in $t in [0,1]$. To keep the exposition transparant, we deliberately exclude the case $H < 1/2$. The expansion is still valid in this case, but the proof requires additional technicalities.

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CWI | |

CWI. Probability, Networks and Algorithms [PNA] | |

Organisation | Stochastics |

Dzhaparidze, K., & van Zanten, H. (2002). A series expansion of fractional Brownian motion with Hurst index exceeding 1/2. CWI. Probability, Networks and Algorithms [PNA]. CWI. |