Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions

This work is devoted to almost sure and moment exponential stability of regime-switching jump diffusions. The Lyapunov function method is used to derive sufficient conditions for stabilities for general nonlinear systems, which further helps to derive easily verifiable conditions for linear systems. For one-dimensional linear regime-switching jump diffusions, necessary and sufficient conditions for almost sure and $p$th moment exponential stabilities are presented. Several examples are provided for illustration.

[1]  D. Applebaum,et al.  Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise , 2009, Journal of Applied Probability.

[2]  Fubao Xi,et al.  Asymptotic properties of jump-diffusion processes with state-dependent switching , 2009 .

[3]  X. Mao.  Stability of stochastic differential equations with Markovian switching , 1999 .

[4]  Gang George Yin,et al.  Stability of Regime-Switching Jump Diffusions , 2010, SIAM J. Control. Optim..

[5]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .

[6]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[7]  G. Barone-Adesi,et al.  Efficient Analytic Approximation of American Option Values , 1987 .

[8]  Qi-Min Zhang,et al.  Stability for multidimensional jump-diffusion processes , 2004, Proceedings of 2004 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.04EX826).

[9]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[10]  Q. Zhang,et al.  Stock Trading: An Optimal Selling Rule , 2001, SIAM J. Control. Optim..

[11]  Gang George Yin,et al.  Almost Sure and pth-Moment Stability and Stabilization of Regime-Switching Jump Diffusion Systems , 2014, SIAM J. Control. Optim..

[12]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[13]  G. Yin,et al.  Hybrid Switching Diffusions: Properties and Applications , 2009 .

[14]  A. Skorokhod.  Asymptotic Methods in the Theory of Stochastic Differential Equations , 2008 .

[15]  Almut E. D. Veraart,et al.  Modelling energy spot prices by volatility modulated Levy-driven Volterra processes , 2013, 1307.6332.

[16]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[17]  E. Seneta.  Fitting the variance-gamma model to financial data , 2004, Journal of Applied Probability.

[18]  Fubao Xi,et al.  Stability and Recurrence of Regime-Switching Diffusion Processes , 2014, SIAM J. Control. Optim..

[19]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[20]  G. Yin,et al.  Stability of regime-switching diffusions , 2007 .

[21]  George Yin,et al.  Feynman–Kac formulas for regime-switching jump diffusions and their applications , 2015, 1702.01495.

[22]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[23]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[24]  Rama Cont,et al.  Integro-differential equations for option prices in exponential Lévy models , 2005, Finance Stochastics.

[25]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .