In this talk, we develop an analytical and computational framework for financial engineering problems in which asset prices are modelled by general one dimensional Markov processes, for example, diffusion processes, jump-diffusion processes, and affine jump diffusions, etc. In particular, we focus on financial pricing problems where the payoffs of financial contracts involve continuous/discrete additive functionals of state variables. Our analysis is based on the infinitesimal generator of the Markov process and we characterize the joint transform of both discrete and continuous additive functionals, and the terminal value. An efficient computational framework is then designed based on a continuous time Markov chain approximation methodology. An exhaustive set of numerical examples illustrate the substantial advantage of our methods. This is a joint work with Zhenyu Cui (Stevens Institute of Technology), Chihoon Lee (Stevens Institute of Technology) and Lingjiong Zhu (Florida State Univeristy).