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doi: 10.3934/dcdsb.2021309
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Exponential ergodicity for regime-switching diffusion processes in total variation norm

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

2. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Fubao Xi

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: The research was supported by the National Natural Science Foundation of China (12071031)

We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.

Citation: Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021309
References:
[1]

J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, Nonlinear Anal. Hybrid Syst., 38 (2020), 100914, 18 pp. doi: 10.1016/j.nahs.2020.100914.  Google Scholar

[2]

J. BaoJ. Shao and C. Yuan, Approximation of invariant measures for regime-switching diffusions, Potential Anal., 44 (2016), 707-727.  doi: 10.1007/s11118-015-9526-x.  Google Scholar

[3]

J.-B. BardetH. Guérin and F. Malrieu, Long time behavior of diffusions with Markov switching, ALEA Lat. Am. J. Probab. Math. Stat., 7 (2010), 151-170.   Google Scholar

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[5]

M.-F. Chen, From Markov Chains to non-Equillibrium Particle Systems, World Scientific Publishing Co. Pte. Ltd, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[6]

M.-F. Chen and S. Li, Coupling methods for multidimensional diffusion processes, Ann. Probab., 17 (1989), 151-177.   Google Scholar

[7]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.  Google Scholar

[8]

B. de Saporta and J.-F. Yao, Tail of linear diffusion with Markov switching, Ann. Appl. Probab., 15 (2005), 992-1018.  doi: 10.1214/105051604000000828.  Google Scholar

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A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New York, 1964.  Google Scholar

[10]

T. Hou and J. Shao, Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching, Sci. China Math., 63 (2020), 1169-1180.  doi: 10.1007/s11425-017-9392-5.  Google Scholar

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Y. HuD. NualartX. Sun and Y. Xie, Smoothness of density for stochastic differential equations with Markovian switching, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3615-3631.  doi: 10.3934/dcdsb.2018307.  Google Scholar

[12]

H. Ji and F. Xi, Stationary distribution of stochastic population dynamics in state-dependent random environments, Systems Control Lett., 144 (2020), 104774, 8 pp. doi: 10.1016/j.sysconle.2020.104774.  Google Scholar

[13]

R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[14]

N. V. Krylov, Introduction to the Theory of Diffusion Processes, American Math. Soc., Providence, RI, 2002. doi: 10.1090/gsm/043.  Google Scholar

[15]

A. Kulik, Ergodic Behavior of Markov Processes, De Gruyter, Berlin, 2018.  Google Scholar

[16]

X. Ma and F. Xi, Large deviations for empirical measures of switching diffusion processes with small parameters, Front. Math. China, 10 (2015), 949-963.  doi: 10.1007/s11464-015-0486-7.  Google Scholar

[17] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.  Google Scholar
[18]

D. H. NguyenG. Yin and C. Zhu, Certain properties related to well posedness of switching diffusions, Stoch. Process. Appl., 127 (2017), 3135-3158.  doi: 10.1016/j.spa.2017.02.004.  Google Scholar

[19]

R. Pinsky and M. Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 519-536.   Google Scholar

[20]

G. O. Roberts and J. S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes, Electron. J. Probab., 1 (1996), 21 pp. doi: 10.1214/EJP.v1-9.  Google Scholar

[21]

J. Shao, Ergodicity of one-dimensional regime-switching diffusion processes, Sci. China Math., 57 (2014), 2407-2414.  doi: 10.1007/s11425-014-4853-8.  Google Scholar

[22]

J. Shao, Ergodicity of regime-switching diffusions in Wasserstein distances, Stoch. Process. Appl., 125 (2015), 739-758.  doi: 10.1016/j.spa.2014.10.007.  Google Scholar

[23]

J. Shao, Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions, SIAM J. Contr. Optim., 56 (2018), 3215-3238.  doi: 10.1137/18M116678X.  Google Scholar

[24]

X. Sun and Y. Xie, Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching, Front. Math. China, 13 (2018), 1447-1467.  doi: 10.1007/s11464-018-0735-7.  Google Scholar

[25]

A. Yu. Veretennikov, Bounds for the mixing rate in the theory of stochastic equations, Theory Probab. Appl., 32 (1988), 273-281.   Google Scholar

[26]

A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stoch. Process. Appl., 70 (1997), 115-127.  doi: 10.1016/S0304-4149(97)00056-2.  Google Scholar

[27]

F. Xi, Invariant measures for a random evolution equation with small perturbations, Acta Math. Appl. Sin. Engl. Ser., 17 (2001), 631-642.  doi: 10.1007/s101140100127.  Google Scholar

[28]

F. Xi, Stability of a random diffusion with nonlinear drift, Stat. Probab. Letters, 68 (2004), 273-286.  doi: 10.1016/j.spl.2004.03.010.  Google Scholar

[29]

F. Xi, Feller property and exponential ergodicity of diffusion processes with state-dependent switching, Sci. China Ser. A, 51 (2008), 329-342.  doi: 10.1007/s11425-007-0147-8.  Google Scholar

[30]

F. Xi and G. Yin, Asymptotic properties of a mean-field model with a continuous-state-dependent switching process, J. Appl. Probab., 46 (2009), 221-243.  doi: 10.1239/jap/1238592126.  Google Scholar

[31]

F. Xi and G. Yin, Jump-diffusions with state-dependent switching: Existence and uniqueness, Feller property, linearization, and uniform ergodicity, Sci. China Math., 54 (2011), 2651-2667.  doi: 10.1007/s11425-011-4281-y.  Google Scholar

[32]

F. Xi and G. Yin, Stochastic Liénard equations with state-dependent switching, Acta Math. Appl. Sin. Engl. Ser., 31 (2015), 893-908.  doi: 10.1007/s10255-015-0538-5.  Google Scholar

[33]

F. Xi, G. Yin and C. Zhu, Regime-switching jump diffusions with non-Lipschitz coefficients and countably many switching states: Existence and uniqueness, Feller, and strong Feller properties, in Modeling, Stochastic Control, Optimization, and Applications (eds. G. Yin and Q. Zhang), Springer, (2019), 571–599.  Google Scholar

[34]

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

[35]

C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stoch. Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2.  Google Scholar

[36]

Z. Zhang, H. Yang, J. Tong and L. Hu, Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching, Stoch. Dyn., 19 (2019), 1950023, 26 pp. doi: 10.1142/S0219493719500230.  Google Scholar

show all references

References:
[1]

J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, Nonlinear Anal. Hybrid Syst., 38 (2020), 100914, 18 pp. doi: 10.1016/j.nahs.2020.100914.  Google Scholar

[2]

J. BaoJ. Shao and C. Yuan, Approximation of invariant measures for regime-switching diffusions, Potential Anal., 44 (2016), 707-727.  doi: 10.1007/s11118-015-9526-x.  Google Scholar

[3]

J.-B. BardetH. Guérin and F. Malrieu, Long time behavior of diffusions with Markov switching, ALEA Lat. Am. J. Probab. Math. Stat., 7 (2010), 151-170.   Google Scholar

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[5]

M.-F. Chen, From Markov Chains to non-Equillibrium Particle Systems, World Scientific Publishing Co. Pte. Ltd, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[6]

M.-F. Chen and S. Li, Coupling methods for multidimensional diffusion processes, Ann. Probab., 17 (1989), 151-177.   Google Scholar

[7]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.  Google Scholar

[8]

B. de Saporta and J.-F. Yao, Tail of linear diffusion with Markov switching, Ann. Appl. Probab., 15 (2005), 992-1018.  doi: 10.1214/105051604000000828.  Google Scholar

[9]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New York, 1964.  Google Scholar

[10]

T. Hou and J. Shao, Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching, Sci. China Math., 63 (2020), 1169-1180.  doi: 10.1007/s11425-017-9392-5.  Google Scholar

[11]

Y. HuD. NualartX. Sun and Y. Xie, Smoothness of density for stochastic differential equations with Markovian switching, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3615-3631.  doi: 10.3934/dcdsb.2018307.  Google Scholar

[12]

H. Ji and F. Xi, Stationary distribution of stochastic population dynamics in state-dependent random environments, Systems Control Lett., 144 (2020), 104774, 8 pp. doi: 10.1016/j.sysconle.2020.104774.  Google Scholar

[13]

R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[14]

N. V. Krylov, Introduction to the Theory of Diffusion Processes, American Math. Soc., Providence, RI, 2002. doi: 10.1090/gsm/043.  Google Scholar

[15]

A. Kulik, Ergodic Behavior of Markov Processes, De Gruyter, Berlin, 2018.  Google Scholar

[16]

X. Ma and F. Xi, Large deviations for empirical measures of switching diffusion processes with small parameters, Front. Math. China, 10 (2015), 949-963.  doi: 10.1007/s11464-015-0486-7.  Google Scholar

[17] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.  Google Scholar
[18]

D. H. NguyenG. Yin and C. Zhu, Certain properties related to well posedness of switching diffusions, Stoch. Process. Appl., 127 (2017), 3135-3158.  doi: 10.1016/j.spa.2017.02.004.  Google Scholar

[19]

R. Pinsky and M. Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 519-536.   Google Scholar

[20]

G. O. Roberts and J. S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes, Electron. J. Probab., 1 (1996), 21 pp. doi: 10.1214/EJP.v1-9.  Google Scholar

[21]

J. Shao, Ergodicity of one-dimensional regime-switching diffusion processes, Sci. China Math., 57 (2014), 2407-2414.  doi: 10.1007/s11425-014-4853-8.  Google Scholar

[22]

J. Shao, Ergodicity of regime-switching diffusions in Wasserstein distances, Stoch. Process. Appl., 125 (2015), 739-758.  doi: 10.1016/j.spa.2014.10.007.  Google Scholar

[23]

J. Shao, Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions, SIAM J. Contr. Optim., 56 (2018), 3215-3238.  doi: 10.1137/18M116678X.  Google Scholar

[24]

X. Sun and Y. Xie, Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching, Front. Math. China, 13 (2018), 1447-1467.  doi: 10.1007/s11464-018-0735-7.  Google Scholar

[25]

A. Yu. Veretennikov, Bounds for the mixing rate in the theory of stochastic equations, Theory Probab. Appl., 32 (1988), 273-281.   Google Scholar

[26]

A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stoch. Process. Appl., 70 (1997), 115-127.  doi: 10.1016/S0304-4149(97)00056-2.  Google Scholar

[27]

F. Xi, Invariant measures for a random evolution equation with small perturbations, Acta Math. Appl. Sin. Engl. Ser., 17 (2001), 631-642.  doi: 10.1007/s101140100127.  Google Scholar

[28]

F. Xi, Stability of a random diffusion with nonlinear drift, Stat. Probab. Letters, 68 (2004), 273-286.  doi: 10.1016/j.spl.2004.03.010.  Google Scholar

[29]

F. Xi, Feller property and exponential ergodicity of diffusion processes with state-dependent switching, Sci. China Ser. A, 51 (2008), 329-342.  doi: 10.1007/s11425-007-0147-8.  Google Scholar

[30]

F. Xi and G. Yin, Asymptotic properties of a mean-field model with a continuous-state-dependent switching process, J. Appl. Probab., 46 (2009), 221-243.  doi: 10.1239/jap/1238592126.  Google Scholar

[31]

F. Xi and G. Yin, Jump-diffusions with state-dependent switching: Existence and uniqueness, Feller property, linearization, and uniform ergodicity, Sci. China Math., 54 (2011), 2651-2667.  doi: 10.1007/s11425-011-4281-y.  Google Scholar

[32]

F. Xi and G. Yin, Stochastic Liénard equations with state-dependent switching, Acta Math. Appl. Sin. Engl. Ser., 31 (2015), 893-908.  doi: 10.1007/s10255-015-0538-5.  Google Scholar

[33]

F. Xi, G. Yin and C. Zhu, Regime-switching jump diffusions with non-Lipschitz coefficients and countably many switching states: Existence and uniqueness, Feller, and strong Feller properties, in Modeling, Stochastic Control, Optimization, and Applications (eds. G. Yin and Q. Zhang), Springer, (2019), 571–599.  Google Scholar

[34]

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

[35]

C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stoch. Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2.  Google Scholar

[36]

Z. Zhang, H. Yang, J. Tong and L. Hu, Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching, Stoch. Dyn., 19 (2019), 1950023, 26 pp. doi: 10.1142/S0219493719500230.  Google Scholar

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