Title: Parametric Inference for Discretely Observed Subordinate Diffusions,
Abstract: Subordinate diffusions are constructed by time changing diffusion processes
with an independent Lévy subordinator. This is a rich family of Markovian
jump processes which exhibit a variety of jump behavior and have found many
applications. This paper studies parametric inference of discretely observed
ergodic subordinate diffusions. We solve the identifiability problem for these
processes using spectral theory and propose a two-step estimation procedure
based on estimating functions. In the first step, we use an estimating function
that only involves diffusion parameters. In the second step, a martingale
estimating function based on eigenvalues and eigenfunctions of the subordinate
diffusion is used to estimate the parameters of the Lévy subordinator and
the problem of how to choose the weighting matrix is solved. When the
eigenpairs do not have analytical expressions, we apply the constant
perturbation method with high order corrections to calculate them numerically
and the martingale estimating function can be computed efficiently. Consistency
and asymptotic normality of our estimator are established considering the
effect of numerical approximation. Through numerical examples, we show that our
method is both computationally and statistically efficient. A subordinate
diffusion model for VIX (CBOE volatility index) is developed which provides
good fit to the data.