Estimating the Hurst index of the solution of a stochastic integral equation

Abstract

Let X(t) be a solution of a stochastic integral equation driven by fractional Brownian motion B^H and let V²_n (X, 2) = \sum^n-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 n^2H−1V_n²(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 V_n²(X, 2) with V_n²(Y, 2), where Y (t) is the Milstein approximation of X(t).