Probabilistic analysis of a class of Markov jump processes

The paper introduces a class of the jump processes. The first compound component represents a Markov jump process with a finite state space. The second compound component jumps synchronously with the first one. Given the first component trajectory, the second component forms a sequence of independent random vectors. The corresponding conditional distributions are known and have intersecting support sets. This makes impossible the exact recovery of the first process component by the second one. The authors prove the Markov property for the considered class of random processes and obtain a collection of their probability characteristics. It includes the infinitesimal generator and its conjugate operator. Their knowledge makes possible the construction of the Kolmogorov equation system describing the evolution of the process probability distribution. Also, a martingale decomposition for an arbitrary function of the considered process was derived. It can be characterized by the solution to a system of linear stochastic differential equations with martingales on the right side. If the functions of the investigated process have finite moments of the second order, one may obtain the quadratic characteristics of martingales.