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OPEN
received: 23 August 2016 accepted: 11 January 2017 Published: 14 February 2017
Visual information and expert’s idea in Hurst index estimation of the fractional Brownian motion using a diffusion type approximation Ali R. Taheriyoun & Meisam Moghimbeygi An approximation of the fractional Brownian motion based on the Ornstein-Uhlenbeck process is used to obtain an asymptotic likelihood function. Two estimators of the Hurst index are then presented in the likelihood approach. The first estimator is produced according to the observed values of the sample path; while the second one employs the likelihood function of the incremental process. We also employ visual roughness of realization to restrict the parameter space and to obtain prior information in Bayesian approach. The methods are then compared with three contemporary estimators and an experimental data set is studied.
General motivation
Fractional Brownian motion (fBm) appears in modeling wide classes of non-stationary stochastic processes. The statistical self-similarity and the ability of to fine-tune the order of Hölder continuity are the famous advantages of this process that make it typically one of the greatest interest in modeling natural phenomena. Typically, the creation of the fBm is attributed to ref. 1 since it investigated the basic properties of fBm and stressed its role in modeling of natural phenomena. However, it had been introduced during the generation of Gaussian spirals in Hilbert space2. Let {B H (t ), t ∈ +} be an fBm that is a Gaussian zero-mean continuous process with stationary increments and the homogeneous stationary incremental variance function σB2H (t ) : = E X (t + s) − X (s) 2 = σ 2 t 2H ,
for any t, s > 0. Consequently, the covariance function is γ H (t , s) : = E [B H (t ) B H (s)] =
σ 2 2H (t + s 2H − t − s 2H ), 2
(1)
where 0 H > 1/2 and t (t − s) H−1/2 1 K H (t , s) = b H − H − s1/2 −H s 2
∫s
t
(u − s)H−1/2 u H−1/2du ,
for 0
OPEN
received: 23 August 2016 accepted: 11 January 2017 Published: 14 February 2017
Visual information and expert’s idea in Hurst index estimation of the fractional Brownian motion using a diffusion type approximation Ali R. Taheriyoun & Meisam Moghimbeygi An approximation of the fractional Brownian motion based on the Ornstein-Uhlenbeck process is used to obtain an asymptotic likelihood function. Two estimators of the Hurst index are then presented in the likelihood approach. The first estimator is produced according to the observed values of the sample path; while the second one employs the likelihood function of the incremental process. We also employ visual roughness of realization to restrict the parameter space and to obtain prior information in Bayesian approach. The methods are then compared with three contemporary estimators and an experimental data set is studied.
General motivation
Fractional Brownian motion (fBm) appears in modeling wide classes of non-stationary stochastic processes. The statistical self-similarity and the ability of to fine-tune the order of Hölder continuity are the famous advantages of this process that make it typically one of the greatest interest in modeling natural phenomena. Typically, the creation of the fBm is attributed to ref. 1 since it investigated the basic properties of fBm and stressed its role in modeling of natural phenomena. However, it had been introduced during the generation of Gaussian spirals in Hilbert space2. Let {B H (t ), t ∈ +} be an fBm that is a Gaussian zero-mean continuous process with stationary increments and the homogeneous stationary incremental variance function σB2H (t ) : = E X (t + s) − X (s) 2 = σ 2 t 2H ,
for any t, s > 0. Consequently, the covariance function is γ H (t , s) : = E [B H (t ) B H (s)] =
σ 2 2H (t + s 2H − t − s 2H ), 2
(1)
where 0 H > 1/2 and t (t − s) H−1/2 1 K H (t , s) = b H − H − s1/2 −H s 2
∫s
t
(u − s)H−1/2 u H−1/2du ,
for 0
