September  2018, 8(3&4): 879-897. doi: 10.3934/mcrf.2018039

Recurrence for switching diffusion with past dependent switching and countable state space

1. 

Department of Mathematics, University of Alabama Tuscaloosa, AL 35401, USA

2. 

Department of Mathematics, Wayne State University, Detroit. MI 48202, USA

In honor of Jiongmin Yong on the occasion of his 60th Birthday

Received  August 2017 Revised  December 2017 Published  September 2018

Fund Project: This research was supported in part by the Air Force Office of Scientific Research under FA9550-15-1-0131. The research of D. Nguyen was also supported by the AMS-Simons Travel grant.

This work continues and substantially extends our recent work on switching diffusions with the switching processes that depend on the past states and that take values in a countable state space. That is, the discrete component of the two-component process takes values in a countably infinite set and its switching rate at current time depends on the value of the continuous component involving past history. This paper focuses on recurrence, positive recurrence, and weak stabilization of such systems. In particular, the paper aims to providing more verifiable conditions on recurrence and positive recurrence and related issues. Assuming that the system is linearizable, it provides feasible conditions focusing on the coefficients of the systems for positive recurrence. Then linear feedback controls for weak stabilization are considered. Some illustrative examples are also given.

Citation: Dang H. Nguyen, George Yin. Recurrence for switching diffusion with past dependent switching and countable state space. Mathematical Control & Related Fields, 2018, 8 (3&4) : 879-897. doi: 10.3934/mcrf.2018039
References:
[1]

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[2]

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M. F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

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R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072.  doi: 10.1016/j.jfa.2010.04.017.  Google Scholar

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N. T. DieuN. H. DuD. H. Nguyen and G. Yin, Protection zones for survival of species in random environment, SIAM J. Appl. Math., 76 (2016), 1382-1402.  doi: 10.1137/15M1032004.  Google Scholar

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N. T. DieuD. H. NguyenN. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dynamic Sys., 15 (2016), 1062-1084.  doi: 10.1137/15M1043315.  Google Scholar

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N. H. Du and D. H. Nguyen, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023.  Google Scholar

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N. H. DuD. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.  Google Scholar

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B. Dupire, Functional Itô's Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS Available at SSRN: http://ssrn.com/abstract=1435551 or http://dx.doi.org/10.2139/ssrn.1435551. Google Scholar

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A. HeningD. Nguyen and G. Yin, Stochastic population growth in spatially heterogeneous environments: The density-dependent case, J. Math. Biology, 76 (2018), 697-754.  doi: 10.1007/s00285-017-1153-2.  Google Scholar

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R. Z. KhasminskiiC. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051.  doi: 10.1016/j.spa.2006.12.001.  Google Scholar

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V. Kulkarni, Fluid models for single buffer systems, in Frontiers in Queueing: Models and Applications in Science and Engineering, 321–338, Ed. J. H. Dashalalow, CRC Press, 1997.  Google Scholar

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D. H. NguyenN. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[20]

D. H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450-2477.  doi: 10.1137/16M1059357.  Google Scholar

[21]

D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Eqs., 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005.  Google Scholar

[22]

D. H. Nguyen and G. Yin, Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space, to appear in Potential Anal., (2017). Google Scholar

[23]

J. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electron. J. Probab., 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018.  Google Scholar

[24]

J. Shao and F. Xi, Strong ergodicity of the regime-switching diffusion processes, Stochastic Process. Appl., 123 (2013), 3903-3918.  doi: 10.1016/j.spa.2013.06.002.  Google Scholar

[25]

J. Shao and F. Xi, Stability and recurrence of regime-switching diffusion processes, SIAM J. Control Optim., 52 (2014), 3496-3516.  doi: 10.1137/140962905.  Google Scholar

[26]

W. M. Wonham, Liapunov criteria for weak stochastic stability, J. Differential Eqs., 2 (1966), 195-207.  doi: 10.1016/0022-0396(66)90043-X.  Google Scholar

[27]

F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818.  doi: 10.1137/16M1087837.  Google Scholar

[28]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[29]

G. Yin, H. Q. Zhang and Q. Zhang, Applications of Two-time-scale Markovian Systems, Science Press, Beijing, China, 2013. Google Scholar

[30]

G. YinG. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382.  doi: 10.1137/110851171.  Google Scholar

[31]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, NY, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[32]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.  doi: 10.1137/060649343.  Google Scholar

[33]

C. Zhu and G. Yin, On strong Feller, recurrence, and weak stabilization of regime-switching diffusions, SIAM J. Control Optim., 48 (2009), 2003-2031.  doi: 10.1137/080712532.  Google Scholar

[34]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and pth-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.  Google Scholar

show all references

References:
[1]

W. J. Anderson, Continuous-time Markov chains: An Applications-Oriented Approach, Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[3]

M. F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[4]

R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072.  doi: 10.1016/j.jfa.2010.04.017.  Google Scholar

[5]

R. Cont and D.-A. Fournié, Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab., 41 (2013), 109-133.  doi: 10.1214/11-AOP721.  Google Scholar

[6]

M. Costa, A piecewise deterministic model for a prey-predator community, Ann. Appl. Probab, 26 (2016), 3491-3530.  doi: 10.1214/16-AAP1182.  Google Scholar

[7]

N. T. DieuN. H. DuD. H. Nguyen and G. Yin, Protection zones for survival of species in random environment, SIAM J. Appl. Math., 76 (2016), 1382-1402.  doi: 10.1137/15M1032004.  Google Scholar

[8]

N. T. DieuD. H. NguyenN. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dynamic Sys., 15 (2016), 1062-1084.  doi: 10.1137/15M1043315.  Google Scholar

[9]

N. H. Du and D. H. Nguyen, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023.  Google Scholar

[10]

N. H. DuD. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.  Google Scholar

[11]

B. Dupire, Functional Itô's Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS Available at SSRN: http://ssrn.com/abstract=1435551 or http://dx.doi.org/10.2139/ssrn.1435551. Google Scholar

[12]

A. HeningD. Nguyen and G. Yin, Stochastic population growth in spatially heterogeneous environments: The density-dependent case, J. Math. Biology, 76 (2018), 697-754.  doi: 10.1007/s00285-017-1153-2.  Google Scholar

[13]

S.-B. HsuT.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosc., 181 (2003), 55-83.  doi: 10.1016/S0025-5564(02)00127-X.  Google Scholar

[14]

R. Z. KhasminskiiC. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051.  doi: 10.1016/j.spa.2006.12.001.  Google Scholar

[15]

V. Kulkarni, Fluid models for single buffer systems, in Frontiers in Queueing: Models and Applications in Science and Engineering, 321–338, Ed. J. H. Dashalalow, CRC Press, 1997.  Google Scholar

[16]

R. F. Luck, Evaluation of natural enemies for biological control: A behavior approach, Trends Ecol. Evol., 5 (1990), 196-199.  doi: 10.1016/0169-5347(90)90210-5.  Google Scholar

[17]

X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood, Chinester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[18]

X. Mao and C. Yuan. Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473.  Google Scholar

[19]

D. H. NguyenN. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[20]

D. H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450-2477.  doi: 10.1137/16M1059357.  Google Scholar

[21]

D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Eqs., 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005.  Google Scholar

[22]

D. H. Nguyen and G. Yin, Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space, to appear in Potential Anal., (2017). Google Scholar

[23]

J. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electron. J. Probab., 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018.  Google Scholar

[24]

J. Shao and F. Xi, Strong ergodicity of the regime-switching diffusion processes, Stochastic Process. Appl., 123 (2013), 3903-3918.  doi: 10.1016/j.spa.2013.06.002.  Google Scholar

[25]

J. Shao and F. Xi, Stability and recurrence of regime-switching diffusion processes, SIAM J. Control Optim., 52 (2014), 3496-3516.  doi: 10.1137/140962905.  Google Scholar

[26]

W. M. Wonham, Liapunov criteria for weak stochastic stability, J. Differential Eqs., 2 (1966), 195-207.  doi: 10.1016/0022-0396(66)90043-X.  Google Scholar

[27]

F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818.  doi: 10.1137/16M1087837.  Google Scholar

[28]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[29]

G. Yin, H. Q. Zhang and Q. Zhang, Applications of Two-time-scale Markovian Systems, Science Press, Beijing, China, 2013. Google Scholar

[30]

G. YinG. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382.  doi: 10.1137/110851171.  Google Scholar

[31]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, NY, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[32]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.  doi: 10.1137/060649343.  Google Scholar

[33]

C. Zhu and G. Yin, On strong Feller, recurrence, and weak stabilization of regime-switching diffusions, SIAM J. Control Optim., 48 (2009), 2003-2031.  doi: 10.1137/080712532.  Google Scholar

[34]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and pth-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.  Google Scholar

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