Using multiple Wiener-Itô stochastic integrals and Malliavin calculus we study the rescaled quadratic variations of a general Hermite process of order q with long-memory (Hurst) parameter H 2 ( 1 2 ; 1). We apply our results to the construction of a strongly consistent estimator for H. It is shown that the estimator is asymptotically non-normal, and converges in the mean-square, after normaliza...

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0, 1]. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wi...

We study a least squares estimator b θT for the Ornstein-Uhlenbeck process, dXt = θXtdt+σdB H t , driven by fractional Brownian motion B H with Hurst parameter H ≥ 1 2 . We prove the strong consistence of b θT (the almost surely convergence of b θT to the true parameter θ). We also obtain the rate of this convergence when 1/2 ≤ H < 3/4, applying a central limit theorem for multiple Wiener integ...

Bridging the gap between an abstract definition of pseudo-differential operators, such as (−1) for − 1 2 < γ < 1 2 , and a concrete way to represent them has proved difficult; deriving stable numerical schemes for such operators is not an easy task either. Thus, the framework of diffusive representations, as already developed for causal fractional integrals and derivatives, is being applied to ...

Let X be a forward diffusion and Y a backward diffusion, both defined on [-0, 1], X t and yt being respectively adapted to the past of a Wiener process W(.), and to its future increments. We construct a "two-sided" stochastic integral of the form. t q~(u, X,, Y") dW(u) 0 which generalizes the backward and forward It6 integrals simultaneously. Our construction is quite intuitive, and leads to a ...

Using multiple Wiener-Itô stochastic integrals and Malliavin calculus we study the rescaled quadratic variations of a general Hermite process of order q with long-memory (Hurst) parameter H 2 ( 1 2 ; 1). We apply our results to the construction of a strongly consistent estimator for H. It is shown that the estimator is asymptotically non-normal, and converges in the mean-square, after normaliza...