Closed-Form Approximations of Moments and Densities of Continuous Time Markov Models

This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump di⁄usions. The proposed expansions extend the ones in Kristensen and Mele (2011) to cover general Markov processes. We demonstrate that the class of expansions nests the transition density and option price expansions developed in Yang et al. (2019) and Wan and Yang (2021) as special cases, thereby connecting seemingly di⁄erent ideas in a uni ed framework. We show how the general expansion can be implemented for fully general jump di⁄usion models. We provide a new theory for the validity of the expansions which shows that series expansions are not guaranteed to converge as more terms are added in general. Thus, these methods should be used with caution. At the same time, the numerical studies in this paper demonstrate good performance of the proposed implementation in practice when a small number of terms are included