Estimating functions for jump-diffusions

TL;DR

This paper extends asymptotic theory for approximate martingale estimating functions to jump-diffusions, establishing conditions for consistency, asymptotic normality, and rate optimality of parameter estimators in high-frequency sampling.

Contribution

It generalizes existing theory to include finite-activity jumps and derives conditions for estimator efficiency and rate optimality for all parameters.

Findings

Estimators are consistent and asymptotically normal.

Drift and jump parameter estimators are rate-optimal.

Additional conditions ensure diffusion parameter rate-optimality and overall efficiency.

Abstract

Asymptotic theory for approximate martingale estimating functions is generalised to diffusions with finite-activity jumps, when the sampling frequency and terminal sampling time go to infinity. Rate optimality and efficiency are of particular concern. Under mild assumptions, it is shown that estimators of drift, diffusion, and jump parameters are consistent and asymptotically normal, as well as rate-optimal for the drift and jump parameters. Additional conditions are derived, which ensure rate-optimality for the diffusion parameter as well as efficiency for all parameters. The findings indicate a potentially fruitful direction for the further development of estimation for jump-diffusions.