International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 347105, 21 pages
Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas
1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive, NW, Calgary, Alberta, T2N 1N4, Canada
2Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, C1A 4P3, Canada
Received 3 August 2010; Accepted 8 November 2010
Academic Editor: Aihua Xia
Copyright © 2010 Anatoliy Swishchuk and M. Shafiqul Islam. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the geometric Markov renewal processes as a model for a security
market and study this processes in a diffusion approximation scheme. Weak convergence
analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion
scheme are presented. We present European call option pricing formulas in the case of
ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.