Convergence in Fractional Models and Applications

Estimating the Hurst Parameter

“…This particular result was shown in Corollary 3.2(i) in Berzin and León [3]. In Section 4.2.1, we shall prove Lemma 3.1.…”

Estimation of the volatility persistence in a discretely observed diffusion model

“…Hence, using the hypothesis of model (5) and Cauchy-Schwarz inequality, we get E[c 2 jk ] ≤ c2 − j (1+2H ) . By a Taylor expansion, we get E[c 0 +· · ·+c r ] ≤ c.…”

“…By Assumption B, this implies Z u = 0 on [β 1 , β 2 ] with positive probability, which is absurd by the assumptions on model (5). Then, for ε > 0, there exists r > 0 such that…”

“…Then, we can recover the Hurst parameter at the parametric rate n −1/2 . Indeed, we can use as follows local properties of the trajectory of the fractional Brownian motion; see Istas and Lang [23], and see also Berzin and Leon [5], Lang and Roueff [25]. Let s = (s 0 , .…”

“…In all the proofs, we repeatedly use the notation c for constants depending on H , ψ and the functions defined in model (1) and (2) or model (5), continuous in their arguments, and that may vary from line to line. We write the symbol = also for almost sure equality and for a function g, we set g ∞ = sup t |g(t)|.…”