arXiv subjects
Publications and source records attributed to Dmytro Ivanenko.
The article considers parameter estimation constructing such as quasi-maximum likelyhood estimation and one step estimation in statistical models generated by solution of stochastic differential equation. It has been developed a software for parameter estimating and has been presented correspondent testing and comparing.
Suppose we have a high-frequency sample from the Lévy process of the form $X_t^θ=βt+γZ_t+U_t$, where $Z$ is a possibly asymmetric locally $α$-stable Lévy process, and $U$ is a nuisance Lévy process less active than $Z$. We prove the LAN property about the explicit parameter $θ=(β,γ)$ under very mild conditions without specific form of the Lévy measure of $Z$, thereby generalizing the LAN result of A\"ıt-Sahalia and Jacod (2007). In particular, it is clarified that a non-diagonal norming may be necessary in the truly asymmetric case. Due to the special nature of the local $α$-stable property, the asymptotic Fisher information matrix takes a clean-cut form.
The LAN property is proved in the statistical model based on discrete-time observations of a solution to a Lévy driven SDE. The proof is based on a general sufficient condition for a statistical model based on a discrete observations of a Markov process to possess the LAN property, and involves substantially the Malliavin calculus-based integral representations for derivatives of log-likelihood of the model.
The difference equations $ξ_{k}=af(ξ_{k-1})+ε_{k}$, where $(ε_k)$ is a square integrable difference martingale, and the differential equation ${\rm d}ξ=-af(ξ){\rm d}t+{\rm d}η$, where $η$ is a square integrable martingale, are considered. A family of estimators depending, besides the sample size $n$ (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of this estimators is investigated.