  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--diffusions. 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, Chen, and Wan (2019) and Wan and
Yang (2021) as special cases, thereby connecting seemingly different ideas in a
unified framework. We show how the general expansion can be implemented for
fully general jump--diffusion 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.
