LAD estimation of locally stable SDE

TL;DR

This paper establishes the asymptotic properties of LAD estimators for locally stable SDEs observed at high frequency, providing explicit objective functions and covering both ergodic and non-ergodic cases.

Contribution

It introduces a fully explicit LAD estimation method for locally stable SDEs, avoiding numerical integration and extending analysis to various sampling regimes.

Findings

Proves asymptotic mixed normality of LAD estimators.

Provides explicit objective functions for efficient computation.

Analyzes both ergodic and non-ergodic cases.

Abstract

We prove the asymptotic mixed normality of the least absolute deviation (LAD) estimator for a locally $\alpha$-stable stochastic differential equation (SDE) observed at high frequency, where $\alpha\in(0,2)$. We investigate both ergodic and non-ergodic cases, where the terminal sampling time diverges or is fixed, respectively, under different sets of assumptions. The objective function for the LAD estimator is expressed in a fully explicit form without necessitating numerical integration, offering a significant computational advantage over the existing non-Gaussian stable quasi-likelihood approach.