  Markov processes are used in a wide range of disciplines, including finance.
The transition densities of these processes are often unknown. However, the
conditional characteristic functions are more likely to be available,
especially for L\'{e}vy-driven processes. We propose an empirical likelihood
approach, for both parameter estimation and model specification testing, based
on the conditional characteristic function for processes with either continuous
or discontinuous sample paths. Theoretical properties of the empirical
likelihood estimator for parameters and a smoothed empirical likelihood ratio
test for a parametric specification of the process are provided. Simulations
and empirical case studies are carried out to confirm the effectiveness of the
proposed estimator and test.
