M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

Abstract

diffusion process with jumps, infinitely many jumps, M-estimation, discrete observation, parametric inference, asymptotic normality, partial efficiency,

Threshold Estimation for Stochastic Processes with Small Noise

“…However, when |z|≤1 ν(dz) = ∞, it is known that the filter 1 {|∆ n k X|≤δn,ǫ} is not enough to separate ∆ n k X's with or without jumps; see Shimizu [26], Lemma 3.3 and some remarks on that. In this section, we shall consider the following ad hoc situation to understand why the filtered LSE looks like asymptotically normal.…”

“…That is, the indicator 1 {|∆ n k X|≤δn,ǫ} plays a role of a filter to split increments with 'large' and 'small' magnitude of shocks; see Shimizu [26], or Shimizu and Yoshida [29] for the fundamental idea of those filters. It would be intuitively clear that if δ n,ǫ → ∞ then θ LSE n,ǫ and θ n,ǫ can be asymptotically equivalent.…”

Quasi-likelihood analysis for the stochastic differential equation with jumps

“…To solve it, they proposed a discrimination filter, which enabled to discriminate asymptotically between increments with jumps and increments without jumps. After that, Shimizu studied M-estimation for the model which Lévy measure has infinite total mass in Shimizu (2006), and nonparametric estimation of density of Lévy measure in Shimizu (2002). It should be noted that Mancini (2004) independently presented consistent estimation of the characteristics of jumps for Poisson-diffusion model.…”

Functional estimation for Lévy measures of semimartingales with Poissonian jumps

“…They judged that a jump had occurred if the absolute size of the increment of neighboring data was larger than a threshold, and approximated the jump size by the corresponding increment. The idea was also applied to the case where infinitely many jumps occur, in Shimizu [25] and Mancini [16].…”