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.

翻译：本文发展了一类广泛矩函数（包括连续时间马尔可夫过程的转移密度和期权价格）的幂级数展开，涵盖跳跃扩散过程。所提出的展开将Kristensen和Mele（2011）的展开推广至一般马尔可夫过程。我们证明此类展开将Yang、Chen和Wan（2019）以及Wan和Yang（2021）发展的转移密度和期权价格展开作为特例包含在内，从而在一个统一框架下连接了看似不同的思想。我们展示了如何对完全一般的跳跃扩散模型实施该通用展开。我们为展开的有效性提供了新的理论依据，表明随着更多项的添加，级数展开一般不能保证收敛。因此，这些方法应谨慎使用。同时，本文的数值研究表明，在仅包含少量项的实际应用中，所提出的实现方法具有良好的性能。