Ergodicity and High-Frequency Inference for Hybrid Switching L\'{e}vy-Driven Stochastic Differential Equations

Hybrid switching Lévy-driven stochastic differential equations with pure-jump noise and state-dependent switching rates are studied under high-frequency observation.

A three-stage inference procedure is proposed for the drift, scale, and switching-rate parameters, combining a staged Gaussian quasi-likelihood with an intensity-type contrast.

Checkable sufficient conditions for weighted exponential ergodicity are established for the hybrid process; the proof does not rely on Brownian smoothing, but uses a fixed skeleton-chain argument combining small-jump accessibility and regime connectivity.

Under ergodicity and the high-frequency sampling scheme, consistency, joint asymptotic normality, and a polynomial-type large deviation inequality are proved for the full estimator.

The joint limit exhibits a transparent covariance structure: the drift and scale blocks are coupled through the third moment of the driving Lévy noise, whereas the switching-rate block is asymptotically uncorrelated with the continuous-coefficient blocks.

Numerical experiments for models driven by normal inverse Gaussian noise illustrate the finite-sample behavior of the proposed estimators.