Markov switching for position dependent random maps with application to forecasting

A Markov switching random map consists of a collection of transformations and a controlling stochastic matrix. In this process, at each time step, one transformation is selected randomly and applied. The selection of the transformations is controlled by the stochastic matrix of the process. In this note, we first prove the existence of absolutely continuous invariant measures (acims) for random maps, whose underlying transformations are piecewise monotonic, controlled by a position dependent stochastic matrix and study the ergodic properties of the acim. In particular, we prove a Birkhoff type ergodic theorem. Then we prove the existence of an acim for another class of Markov switching random maps based on geometric properties of the underlying transformations. We apply these results to forecasting in financial markets. © 2005 Society for Industrial and Applied Mathematics.