Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind

Abstract

Fractional Ornstein-Uhlenbeck process of the second kind (fOU 2 ) is solution of the Langevin equation dX t = −θX t dt + dY . Then using the ergodicity of fOU 2 process, we construct consistent estimators of drift parameter θ based on discrete observations in two possible cases: (i) the Hurst parameter H is known and (ii) the Hurst parameter H is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range H ∈ ( 1 2 , 1).

Generalized k-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

“…which implies that for large N we have approximately π a (2) , N H (0) ∼ 2 2H π a (1) , N H (0). This, in turn, can be transferred to S N :…”

“…Assume that the time t > 1 at which the solution (3) is observed is not known. Similarly to [10], if two sequences (a (1) i ) i∈{0,...,p} and (a (2) i ) i∈{0,...,2p} are considered, where a (2) is obtained by "thinning" the sequence a (1) (i.e., a (2) 2k := a (1) k for k ∈ {0, . .…”

“…Nowadays, a particular case of wide interest is represented by the S(P)DE driven by fractional Brownian motion (fBm) and related processes, due to the vast area of application of such stochastic models. Many recent works concern the estimation of the drift parameter for stochastic equation driven by fractional Brownian motion (we refer, among many others, to [1], [9], [12], [24]), while fewer works deal with the estimation of the Hurst parameter in such stochastic equations.…”

AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation

“…Then Q n > 2σ 2 − σ 2 > 0, therefore Q n is bounded away from 0 and the term |Q n | −5p /2 has no singularity for any p > 1. For the term |Q n − 2σ 2 | −3p /2 , we put C := µ−2σ 2 2 , the constant C = 0, because µ = 2σ 2 ⇔ a 1 = 0, we can assume a 1 = 0, because there is no AR(1) process with a 1 = 0. Therefore, we can pick n such that 1 √ n < C. In this case Q n − 2σ 2 = Q n − µ + 2C > − 1 √ n + 2C > 2C − C > 0, hence the term |Q n − 2σ 2 | −3p /2 has no singularities at 2σ 2 for any p > 1.…”

Optimal rates for parameter estimation of stationary Gaussian processes

“…which finishes the lower bound of the theorem.The upper bound is easier to prove, and follows from the same estimates as for the lower bound. Details are omitted.Remark 16The non-central limit theorem in Remark 7 part (1) also holds if Z is replaced by Z +Y under assumption (24) if γ > 1/2 ; and similarly for part(2) if γ > H − (q − 1) /2q. These results' proofs, which are omitted, follow the results in Remark 7 and from the tools in this section and those in[20] and[5].…”