Estimating the Hurst index of the solution of a stochastic integral equation - Kubilius, Kęstutis; Melichov, Dmitrij (Lietuvos matematikos rinkinys, 2009)
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(δ k2X)2 be the second order quadratic variation, where δ 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).
Keywords: fractional Brownian motion; quadratic variation; consistent estimator; Milstein approximation