Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

Berzin, Corinne; Latour, Alain; León, J. B.

doi:10.1007/978-3-319-07875-5

Cited by 44 publications

(35 citation statements)

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“…The diversity of properties makes the fBm useful in modeling (see e.g. [7]), various aspects of the related theory and applications can be found in [20], [8], [36], [41], [50]. The two main operators of interest in the fractional setting are (Kf )(t) = The first one is the covariance operator of the fBm itself and the second operator corresponds to the fractional Brownian noise, that is, the formal derivative of the fBm.…”

Section: Introductionmentioning

confidence: 99%

Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Chigansky

Kleptsyna

2018

Stochastic Processes and their Applications

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Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of L 2 -small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.

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Section: Introductionmentioning

confidence: 99%

Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Chigansky

Kleptsyna

2018

Stochastic Processes and their Applications

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“…There exist several other methods of the Hurst index evaluation based on power variations of X. We refer to the books [5,20] for further information on this subject.…”

Section: Preliminariesmentioning

confidence: 99%

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

Lohvinenko¹,

Ralchenko²

2019

Modern Stochastics: Theory and Applications

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We investigate the fractional Vasicek model described by the stochastic differential equation dXt = (α − βXt) dt + γ dB H t , X0 = x0, driven by the fractional Brownian motion B H with the known Hurst parameter H ∈ (1/2, 1). We study the maximum likelihood estimators for unknown parameters α and β in the non-ergodic case (when β < 0) for arbitrary x0 ∈ R, generalizing the result of Tanaka, Xiao and Yu (2019) for particular x0 = α/β, derive their asymptotic distributions and prove their asymptotic independence.

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“…The estimators H (i) n , i = 1, 2, 3, 4, were considered in [13], [12], [3], and [1]. The estimator H (2) n can be used to estimate the Hurst index of the generic form of the SDE with an additional restriction on the diffusion coefficient.…”

Section: Remarkmentioning

confidence: 99%