Limit behavior of the Rosenblatt Ornstein–Uhlenbeck process with respect to the Hurst index

Abstract

We study the convergence in distribution, as H → 1 2 and as H → 1, of the integral R f (u)dZ H (u), where Z H is a Rosenblatt process with self-similarity index H ∈ 1 2 , 1 and f is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.2010 AMS Classification Numbers: 60H05, 60H15, 60G22.

Behavior with respect to the Hurst index of the Wiener Hermite integrals and application to SPDEs

“…The diffrence to the non-stationary case is that the function f from the last proof has support of infinite Lebesque measure an we need to use an argument based on the power counting theorem when H tends to one half. The proof of this results is similar in spirit to the proofs of Proposition 6 and Proposition 7 in [22].…”

“…Notice that q = 2 and d = 1 we retrieve the results in [22]. For f = 1, the results in this section reduces to those in Theorem 1 from [1].…”

“…Notice that in this case the space H A k given by (15) coincides with L 1 (R). Since it is clear that f belongs to |H H | ∩ L 1 (R) (see [22]), we get immediatly by Proposition 1 the convergence as…”

“…The proof is completed by showing that (31) is satisfied. We have We recall (see [22]) that for p ≥ 1,…”

“…Our work continues a recent line of research that concerns the limit behavior in distribution with respect to the Hurst parameter of Hermite and related fractionaltype stochastic processes. In particular, the papers [5] and [2] deal with the asymptotic behavior of the generalized Rosenblatt process, the work [1] studies the multiparamter Hermite processes while the paper [22] investigates the Ornstein-Uhlenbeck process with Hermite noise of order q = 2.…”