LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

Abstract

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al. [Preprint HAL-01410989v2 (2017)].

Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

“…We proved that the LAMN property holds (Corollary 2.4 in [6]) for the parameter β with rate r n and information matrix I = I 11 0 0 I 22 where…”

“…The uniform convergence with respect to x 0 is not required in this paper but will be useful in [6].…”

“…Considering the statistical experiment (R n , B n , P β n ) corresponding to the observation of (X β i/n ) 1≤i≤n , Theorems 2.1 and 2.2 permit to prove in [6] the L 2 -regularity property of the transition density p β 1/n (x, y):…”

“…) (see Theorem 2.1 in [6]) and to deduce the Local Asymptotic Mixed Normality property (see Jeganathan [12]). We proved that the LAMN property holds (Corollary 2.4 in [6]) for the parameter β with rate r n and information matrix I = I 11 0 0 I 22 where…”

“…Indeed, considering the estimation of β from the discrete time observations (X β i/n ) 0≤i≤n , and denoting by p β 1/n (x, y) the transition density 1 of the discrete time process, the estimation rate of the parameter β strongly relies on the asymptotic behavior of the derivative ∇ β p β 1/n (x, y), as n goes to infinity. Based on the results established in the present paper, we derive, in [6], an asymptotic expansion of the log-likelihood ratio and we prove the LAMN property for the parameter β.…”