Estimating the Hurst index of the solution of a stochastic integral equation
Articles
Kęstutis Kubilius
Vilnius Gediminas Technical University
Dmitrij Melichov
Vilnius Gediminas Technical University
Published 2009-12-20
https://doi.org/10.15388/LMR.2009.04
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Keywords

fractional Brownian motion
quadratic variation
consistent estimator
Milstein approximation

How to Cite

Kubilius, K. and Melichov, D. (2009) “Estimating the Hurst index of the solution of a stochastic integral equation”, Lietuvos matematikos rinkinys, 50(proc. LMS), pp. 24–29. doi:10.15388/LMR.2009.04.

Abstract

Let X(t) be a solution of a stochastic integral equation driven by fractional Brownian motion BH and let V2n (X, 2) = \sumn-1 k=1(\delta k2X)2 be the second order quadratic variation, where \delta k2X = X (k+1/N) − 2X (k/ n) +X (k−1/n). Conditions under which n2H−1Vn2(X, 2) converges almost surely as n → ∞ was obtained. This fact is used to get a strongly consistent estimator of the Hurst index H, 1/2 < H < 1. Also we show that this estimator retains its properties if we replace Vn2(X, 2) with Vn2(Y, 2), where Y (t) is the Milstein approximation of X(t).

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