On the Numerical Computation of Transition Matrix pth Root

Transition matrices arise in a wide class of engineering problems especially in mathematical models using Markov chains. Usually they describe the transition probabilities of a vector state at time $t$ to the same state vector at time $t+\Delta t$, where $\Delta t$ is the shortest period over which a transition matrix can be estimated. If a short term transition matrix is needed it can be obtained by computing a $p$th root. For example in risk management of portfolio the company's credit ratings are recorded yearly, thus to define, at the end of the year, a transition matrix with all the recorded information. However, investment horizon is shorter than a year, thus to require the computation of the matrix $p$th root. In this paper we consider the numerical computation of the $p$th root of a transition matrix. The aim of the paper is to highlight the properties of some numerical methods preserving the geometric peculiarities of the pth root of a transition matrix.