The Geometric Markov Renewal Processes with Application to Finance

Abstract

We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.

Keywords:

• Averaged and double averaged GMRP
• Geometric Markov renewal process (GMRP)
• Martingale properties of GMRP
• Merged GMRP
• Random evolutions
• Rates of convergence

Mathematics Subject Classification:

• Primary 60K15, 91G80
• Secondary 60F37H, 60G42

Acknowledgments

This research is partially supported by the University of Prince Edward Island major research grants (MRG) of M. S. Islam and NSERC grant of A. Swishchuk.