Abstract
We study the minimum Skorohod distance estimation theta _{varepsilon}^{ast } and minimum L_{1}-norm estimation widetilde {theta _{varepsilon}} of the drift parameter θ of a stochastic differential equation dX_{t}=theta X_{t},dt+varepsilon ,dL^{d}_{t}, X_{0}=x_{0}, where {L^{d}_{t},0leq tleq T} is a fractional Lévy process, varepsilon in (0,1]. We obtain their consistency and limit distribution for fixed T, when varepsilon rightarrow 0. Moreover, we also study the asymptotic laws of their limit distributions for Trightarrow infty .
Highlights
Statistical inference for stochastic equations is a main research direction in probability theory and its applications
Genon-Catalot [8] and Laredo [17] considered the efficient estimation for drift parameters of small diffusions from discrete observations as → 0 and n → ∞
Using a contrast function under suitable conditions on and n, Sørensen and Uchida [28] and Gloter and Sørensen [9] considered the efficient estimation for unknown parameters in both drift and diffusion coefficient functions
Summary
Statistical inference for stochastic equations is a main research direction in probability theory and its applications. Shen and Yu [26] obtained consistency and the asymptotic distribution of the estimator for Ornstein–Uhlenbeck processes with small fractional Lévy noises. Prakasa Rao [25] studied the minimum L1-norm estimates of the drift parameter of Ornstein–Uhlenbeck process driven by fractional Brownian motion and investigated the asymptotic properties following Kutoyants and Pilibossian [14, 15]. Motivated by the above results, in this paper we consider the minimum Skorohod distance estimation θε∗ and minimum L1-norm estimation θε of the drift parameter θ for Ornstein–Uhlenbeck processes driven by the fractional Lévy process {Ldt , 0 ≤ t ≤ T} which satisfies the following stochastic differential equation: dXt = θ Xt dt + dLdt , X0 = x0,.
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