Transform Analysis for Markov Processes and Applications: An Operator-based Approach

Abstract

We establish a novel duality relationship between continuous and discrete non-negative additive functionals of stochastic (not necessarily Markovian) processes and their right inverses. For general Markov processes, we further extend and develop a theoretical and computational framework for the transform analysis via an operator-based approach, i.e. through the infinitestimal generators. More precisely, we characterize the joint double transforms of additive functionals of Markov processes and the terminal values in both discrete and continuous time. In particular, under the continuous-time Markov chain (CTMC) setting, we obtain single Laplace transforms for continuous/discrete additive functionals and their inverses. Lastly, we discuss the potential applications of the proposed transform methodology in various contextual areas in finance and queuing theory within operations research.