We prove the asymptotic mixed normality of the least absolute deviation (LAD) estimator for a locally $α$-stable stochastic differential equation (SDE) observed at high frequency, where $α\in(0,2)$. 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 signi
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