LAD estimation of locally stable SDE

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.