Analysis and Estimation of the States of Special Jump Markov Processes. I. Martingale Representation

The first part of this paper was devoted to a class of continuous-time jump processes generalizing the finite-state Markov processes. Main characteristics of this process such as the transition probabilities, infinitesimal generator, and so on were established. Processes of this class were proved to be solutions of linear differential equations with a martingale in the right-hand side. Stochastic analysis of a hidden Markov model of evolution of risky assets was presented as an example.