Two-stage semiparametric inference for regime-switching jump diffusions with unknown L\'evy densities

We study high-frequency semiparametric inference for ergodic regime-switching jump diffusions whose continuous coefficients are parametric and whose regime-wise Lévy 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 Lévy 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.