Convergence, boundedness, and ergodicity of regime-switching diusion processes with infinite memory

Abstract

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.

Keywords

Regime-switching diffusion process / infinite memory / convergence / boundedness / Feller property / invariant measure / Wasserstein distance

References

[1] Bao J, Shao J, Yuan C. Approximation of invariant measures for regime-switching diffusions.Potential Anal, 2016, 44: 707–727

[2] Bao J, Shao J, Yuan C. Invariant measures for path-dependent random diffusions. arXiv: 1706.05638v1

[3] Bao J, Wang F-Y, Yuan C. Ergodicity for neutral type SDEs with infinite length of memory. arXiv: 1805.03431v3

[4] Bao J, Yin G,Yuan C. Ergodicity for functional stochastic differential equations and applications. Nonlinear Anal, 2014, 98: 66–82

[5] Bardet J, Guérin H, Malrieu F. Long time behavior of diffusions with Markov switching. ALEA Lat Am J Probab Math Stat, 2010, 7: 151–170

[6] Chen M-F. From Markov Chains to Non-equilibrium Particle Systems.Singapore: World Scientific Publishing Co Pte Ltd, 2004

[7] Cloez B, Hairer M. Exponential ergodicity for Markov processes with random switching. Bernoulli, 2015, 21: 505–536

[8] Da Prato G, Zabczyk J. Ergodicity for Infinite Dimensional Systems.Cambridge: Cambridge Univ Press, 1996

[9] Hairer M, Mattingly J C, Scheutzow M. Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations. Probab Theory Related Fields, 2011, 149: 223–259

[10] Hale J K, Lunel S M V. Introduction to Functional Differential Equations.Berlin: Springer-Verlag, 1993

[11] Hino Y, Naito T, Murakami S. Functional Differential Equations with Infinite Delay. Berlin-New York: Springer-Verlag, 1991

[12] Kappel F, Schappacher W. Some considerations to the fundamental theory of infinite delay equation. J Differential Equations, 1980, 37: 141–183

[13] Ma X, Xi F. Large deviations for empirical measures of switching diffusion processes with small parameters. Front Math China, 2015, 10: 949–963

[14] Mao X. Stochastic Differential Equations and Applications.Chichester: Horwood, 1997

[15] Mao X, Yuan C. Stochastic Differential Equations with Markovian Switching. London:Imperial College Press, 2006

[16] Mohammed S-E A. Stochastic Functional Differential Equations. Harlow-New York: Longman, 1986

[17] ksendal B.Stochastic Differential Equations: An Introduction with Applications. 6th ed. Berlin: Springer-Verlag, 2003

[18] Shao J. Ergodicity of regime-switching diffusions in Wasserstein distances. Stochastic Process Appl, 2015, 125: 739–758

[19] Shao J. Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions. SIAM J Control Optim, 2018, 56: 3215–3238

[20] Tong J, Jin X, Zhang Z.Exponential ergodicity for SDEs driven by ff-stable processes with Markovian switching in Wasserstein distances. Potential Anal, 2017, 97: 1–22

[21] Villani C. Optimal Transport: Old and New. Grundlehren Math Wiss, Vol 338.Berlin- Heidelberg: Springer, 2009

[22] Wang L, Wu F.Existence, uniqueness and asymptotic properties of a class of nonlinear stochastic differential delay equations with Markovian switching. Stoch Dyn, 2009, 9: 253–275

[23] Wu F,Xu Y. Stochastic Lotka-Volterra population dynamics with infinite delay. SIAM J Appl Math, 2009, 70: 641–657

[24] Wu F, Yin G, Mei H. Stochastic functional differential equations with infinite delay: existence and uniqueness of solutions, solution map, Markov properties, and ergodicity. J Differential Equations, 2017, 262: 1226–1252

[25] Xi F. On the stability of jump-diffusions with Markovian switching. J Math Anal Appl, 2008, 341: 588–600

[26] Xi F. Asymptotic properties of jump-diffusion processes with state-dependent switching. Stochastic Process Appl, 2009, 119: 2198–2221

[27] Xi F, Zhu C. On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes. SIAM J Control Optim, 2017, 55: 1789–1818

[28] Yin G, Xi F. Stability of regime-switching jump diffusions. SIAM J Control Optim, 2010, 48: 4525–4549

[29] Yin G, Zhu C.Hybrid Switching Diffusions: Properties and Applications.New York: Springer, 2010

[30] Yuan C, Zou J, Mao X. Stability in distribution of stochastic differential delay equations with Markovian switching. Systems Control Lett, 2003, 50: 195–207

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Higher Education Press