Efficient estimation of jump parameters for stochastic differential equations driven by L{\'e}vy processes

Abstract

In a high-frequency context, we investigate the efficient estimation of scaling and jump activity parameters for a stochastic differential equation driven by a L{\'e}vy process with both diffusion component and pure-jump component. We first study efficiency for the prototype L{\'e}vy process. With an in-depth analysis of the behavior of the density of the process in small time, we prove that the LAN Property holds for the joint estimation of the diffusion, scaling and jump activity parameters. We next consider a stochastic equation driven both by a Brownian Motion and a locally stable pure-jump L{\'e}vy process. Using a quasi-likelihood estimation method, we exhibit an estimator that attains the optimal rate of convergence previously identified. The asymptotic properties of the estimator are derived from sharp approximation results for the stochastic equation and for the locally stable distribution.