Two-stage semiparametric inference for regime-switching jump diffusions with unknown Lévy densities

We study high-frequency semiparametric inference for ergodic regime-switching jump diffusions whose continuous coefficients are parametric and whose regime-wise L\'evy densities are unknown. The motivation is that jumps contaminate increments while their law is itself unknown, making likelihood-based inference circular in switching models. We propose a two-stage procedure. First, small increments are used in a truncated Gaussian quasi-likelihood to estimate the drift and diffusion parameters. Second, large drift-corrected residuals are sorted by regime and smoothed with a kernel, with normalization by empirical regime exposure time, to estimate the L\'evy intensity densities on compact sets away from zero. We establish consistency and mixed-rate asymptotic normality for the quasi-maximum likelihood estimator, and derive $L^2(B)$-convergence rates for the exposure-normalized residual density estimator. Simulations for switching Ornstein--Uhlenbeck models illustrate the finite-sample performance of the method.