Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case

摘要

In this article a numerical solution for the evolution equation of a continuous time non-homogeneous semi-Markov process (NHSMP) is obtained using a quadrature method. The paper, after a short introduction to continuous time NHSMP, presents the numerical solution of the process evolution equation with a general quadrature method. Furthermore, the paper gives results that justify this approach, proving that the numerical solution tends to the evolution equation of the continuous time NHSMP. Moreover, the formulae related to some specific quadrature methods are given and a method for obtaining the discrete time NHSMP by applying a very particular quadrature formula for the discretization is shown. In this way the relation between the continuous and discrete time NHSMP is proved. Then, the problem of obtaining the continuous time NHSMP from the discrete one is considered. This problem is solved showing that the discrete process converges in law to the continuous one if the discretized time interval tends to zero. In addition, the discrete time NHSMP in matrix form is presented, and the fact that the solution to this process always exists is proved. Finally, an algorithm for solving the discrete time NHSMP is given. To illustrate the use of this algorithm for a discrete NHSMP, an example in the area of finance is presented.q