Periodic solutions of hybrid jump diffusion processes

Abstract

We investigate periodic solutions of regime-switching jump diffusions. We first show the well-posedness of solutions to stochastic differential equations corresponding to the hybrid system. Then, we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups. Finally, we establish the existence and uniqueness of periodic solutions. Concrete examples are presented to illustrate the results.

Keywords

Hybrid system / regime-switching jump diffusion / periodic solution / strong Feller property / irreducibility

References

[1] Chen L, Dong Z, Jiang J, Zhai J. On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity. Sci China Math, 2020, 63: 1463–1504

[2] Chen X, Chen Z, Tran K, Yin G. Properties of switching jump diffusions: maximum principles and Harnack inequalities. Bernoulli, 2019, 25: 1045–1075

[3] Chen X, Chen Z, Tran K, Yin G. Recurrence and ergodicity for a class of regime-switching jump diffusions. Appl Math Optim, 2019, 80: 415–445

[4] Guo X X, Sun W. Periodic solutions of stochastic differential equations driven by Lévy noises. J Nonlinear Sci, 2021, 31: 32

[5] Hu H, Xu L. Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations. J Math Anal Appl, 2018, 466: 896–926

[6] Khasminskii R Z. Stochastic Stability of Differential Equations. 2nd ed. Berlin: Springer-Verlag, 2012

[7] Lorenz E N. Deterministic nonperiodic ow. J Atmos Sci, 1963, 20: 130–141

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

[9] Meyn S P, Tweedie R L. Markov Chains and Stochastic Stability. Comm Control Engrg Ser. London: Springer-Verlag, 1993

[10] Nguyen D H, Yin G. Modeling and analysis of switching diffusion systems: past-dependent switching with a countable state space. SIAM J Control Optim, 2016, 54: 2450–2477

[11] Nguyen D H, Yin G. Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space. Potential Anal, 2018, 48: 405–435

[12] Shao J. Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space. SIAM J Control Optim, 2015, 4: 2462–2479

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

[14] Xi F, Yin G. Jump-diffusions with state-dependent switching: existence and uniqueness, Feller property, linearization, and uniform ergodicity. Sci China Math, 2011, 12: 2651–2667

[15] Xi F, Yin G, Zhu C. Regime-switching jump diffusions with non-Lipschitz coefficients and countably many switching states: existence and uniqueness, Feller, and strong Feller properties. In: Yin G, Zhang Q, eds. Modeling, Stochastic Control, Optimization, and Applications. IMA Vol Math Appl, Vol 164. Cham:Springer, 2019, 571–599

[16] 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

[17] Yin G, Zhu C. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability, Vol 63.New York: Springer, 2010

[18] Zhang X,Wang K, Li D. Stochastic periodic solutions of stochastic differential equations driven by Lévy process. J Math Anal Appl, 2015, 430: 231–242

RIGHTS & PERMISSIONS

2021 Higher Education Press