Position Dependent Random Maps

In the previous chapter, we presented random maps where the probabilities of switching from one map to another in the process of iteration are constants. In this chapter, we present more general random maps where the probabilities of switching from one map to another are position dependent. Random maps with position dependent probabilities provide a useful framework for analyzing various physical, engineering, social, and economic phenomena. There are useful techniques in the theory of position dependent random maps which can be implemented in finance. For example, position dependent random maps are mathematical models for generalized binomial trees in finance. In Chapter 3, we have presented random maps with constant probabilities for the evolution of financial securities where both up move factor and down move factor are constants and they do not depend on the current values of the securities. In this chapter, we show that position dependent random map models are more general models and we can study financial securities where both up move factor and down move factor depend on the current values of the securities. The density functions of invariant measures for position dependent random maps are useful tools for the study of long term statistical behavior of some financial securities. Position dependent random maps for piecewise C2 expanding maps were first introduced by Go´ra and Boyarsky in [14]. Bahsoun and Go´ra [2] proved the existence of invariant measures for position dependent random maps under milder conditions. Islam, Go´ra, and Boyarsky [17] proved the necessary and sufficient conditions for a general class of position dependent random maps. Markov switching position dependent random maps in one dimension was presented in [3, 18]. For higher dimensional Markov switching position dependent random maps the existence of invariant measures was studied by Islam in [15]. In this chapter, we present the existence of invariant measures, methods for the approximation of densities of absolutely continuous invariant measures for position dependent random maps. Some applications of position dependent random maps in finance are also presented. Our presentation is based on [2-7, 13-18].