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

Parameter estimation and model testing for Markov processes via conditional characteristic functions