On the Action of a Semi-Markov Process on a System of Differential Equations

We deal with a model equation for stochastic processes that results from the action of a semi-Markov process on a system of ordinary differential equations. The resulting stochastic process is deterministic in pieces, with random changes of the motion at random time epochs. By using classical methods of probability calculus, we first build and discuss the fundamental equation for the statistical analysis, i.e. a Liouville Master Equation for the distribution functions, that is a system of hyperbolic PDE with non-local boundary conditions. Then, as the main contribute to this paper, by using the characteristics' method we recast it to a system of Volterra integral equations with space fluxes, and prove existence and uniqueness of the solution. A numerical experiment for a case of practical application is performed.