Filtering and the EM-Algorithm for the Markovian Arrival Process

In this article, we deal with the so-called Markovian Arrival Process (MAP). An MAP is thought of as a partially observed Markov process, so that the Expectation-Maximization (EM) algorithm is a natural way to estimate its parameters. Then, nonlinear filters of basic statistics related to the MAP must be computed. The forward–backward principle is the basic way to do it. Here, bearing in mind a filter-based formulation of the EM-algorithm proposed by Elliott, these filters are shown to be the solution of nonlinear stochastic differential equations (SDEs) which allows a recursive computation. This is well suited for processing large data sets. We also derive linear SDEs or Zakai equations for the so-called unnormalized filters.