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International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 185474, 17 pages
http://dx.doi.org/10.1155/2012/185474
Research Article

Asymptotic Stability of Semi-Markov Modulated Jump Diffusions

Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

Received 28 February 2012; Revised 6 July 2012; Accepted 12 July 2012

Academic Editor: Lukasz Stettner

Copyright © 2012 Amogh Deshpande. 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 class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator. We find conditions under which the solutions of this class of switching jump-diffusion processes are almost surely exponentially stable and moment exponentially stable. We also provide conditions that imply almost sure convergence of the trivial solution when the moment exponential stability of the trivial solution is guaranteed. We further investigate and determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equations 𝑑 𝑋 𝑡 / 𝑑 𝑡 = 𝑓 ( 𝑋 𝑡 ) is almost surely exponentially stable. It is observed that for a one-dimensional state space, a linear unstable system of differential equations when stabilized just by the addition of the jump part of an sMMJD process does not get destabilized by any addition of a Brownian motion. However, in a state space of dimension at least two, we show that a corresponding nonlinear system of differential equations stabilized by jumps gets destabilized by addition of Brownian motion.