Local estimation of the Hurst index of multifractional Brownian motion by increment ratio statistic method

Abstract

We investigate here the Central Limit Theorem of the Increment Ratio Statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer-Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.

A general class of multifractional processes and stock price informativeness

“…There are so far a number of estimation strategies existing in literature. We refer to [17,18,10,5,26,39] and the references therein.…”

“…Coeurjolly [17,18] estimates the PHE of an mBm, starting from an observed discrete sample path of that mBm, using the LGQV approach (see also [16]). Bertrand et al [10] study the same estimation problem as in [17,18], using the nonparametric estimation approach -increment ratio (IR) statistic method.…”

Estimation of the pointwise Hölder exponent of hidden multifractional Brownian motion using wavelet coefficients

“…To this end we take Φ 1 (x) = x to let Y be an mBm. By using the MATLAB functions VariaIR MBM (eta,n,tot) and HestIR(R,p) obtained from http://samm.univ-paris1.fr/-Jean-Marc-Bardet, we also provide the empirical mean and standard deviation (based on 100 independent scenarios of mBm and α = 0.5) of IR2 estimators as follows: Table 2: Mean and std of the IR2 estimators with n = 2 13 and t 0 = 0.1, 0.3, 0.5, 0.7, 0.9.…”

“…Note that in the function VariaIR MBM(eta,n,tot), the mBm was previously generated using the Choleski decomposition of the covariance matrix, so that the sample size n is limited to 6000. However here we have generated the mBm using Wood & Chan circulant matrix, some krigging and a prequantification (see Chan and Wood 1998;Barrière 2007), the sample size n can be thus taken as 2 13 = 8192. The empirical comparison shows no significant difference between the performances of H Y,2 Jn ,n (t 0 ) and H IR2 n,α (t 0 ), except that IR2 estimator has less variance.…”

“…We refer to Rosenbaum (2008), Coeurjolly (2005Coeurjolly ( , 2006, Bardet (2002), Bardet and Surgailis (2013) and Bertrand et al (2013). Rosenbaum (2008) and Bardet (2002) used wavelet-based methods to study inference problem on fBm, i.e., the Hurst parameter is constant; Coeurjolly (2005Coeurjolly ( , 2006 and Bertrand et al (2013) studied the estimation of H(t 0 ) by using respectively generalized variations (see also Chan et al 1995) and increment ratio statistic method, where a discretized sample path of the mBm is assumed to be observed. Bardet and Surgailis (2013) developed a nonparametric estimation method (based on the increment ratio) for evaluating the local Hurst function of a multifractional Gaussian process (whose increments are asymptotically a multiple of an fBm) which extends mBm, starting from a discrete sample path of this process.…”