International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 347105, 21 pages
doi:10.1155/2010/347105
Research Article

Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas

Anatoliy Swishchuk1 and M. Shafiqul Islam2

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.

Abstract

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.