Estimating functions for jump-diffusions

The theory of approximate martingale estimating functions for continuous diffusions is well developed and encompasses many estimators proposed in the literature. This paper extends the asymptotic theory for approximate martingale estimating functions to diffusions with finite-activity jumps. The primary aim is to shed light on the question of rate optimality and efficiency of estimators when observations of a jump-diffusion process are made at increasing frequency, with terminal sampling time going to infinity. Under mild assumptions, it is shown that approximate martingale estimating functions yield consistent and asymptotically normal estimators in the presence of jumps, and a consistent estimator of the asymptotic variance is provided. The estimators are rate optimal for parameters of the drift and jump components of the process. Additional conditions are derived, under which estimators of a diffusion coefficient parameter are rate optimal and therefore converge at a faster rate. These are supplemented with conditions ensuring efficiency of the estimators. Interestingly, the efficiency conditions for jump parameters are much more restrictive than for parameters of the drift and diffusion coefficients. The conditions for both rate optimality and efficiency are, in the established framework of approximate martingale estimating functions, very restrictive. However, these conditions contribute valuable insight into the characteristics of asymptotically well-performing estimating functions, and thus indicate a potentially fruitful direction for further development of estimation for diffusions with jumps.