Periodic solutions of hybrid jump diffusion processes

doi:10.1007/s11464-021-0937-2

Front. Math. China

2021, Vol. 16

Issue (3) : 705-725 https://doi.org/10.1007/s11464-021-0937-2

RESEARCH ARTICLE

Periodic solutions of hybrid jump diffusion processes

Xiaoxia GUO^1,², Wei SUN³(

)

¹. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
². School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
³. Department of Mathematics and Statistics, Concordia University, Montreal H3G 1M8, Canada

Download: PDF(333 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

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

Corresponding Author(s): Wei SUN

Issue Date: 14 July 2021

Cite this article:

Xiaoxia GUO,Wei SUN. Periodic solutions of hybrid jump diffusion processes[J]. Front. Math. China, 2021, 16(3): 705-725.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0937-2
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I3/705

1	L Chen, Z Dong, J Jiang, J Zhai. On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity. Sci China Math, 2020, 63: 1463–1504 https://doi.org/10.1007/s11425-018-9527-1
2	X Chen, Z Chen, K Tran, G Yin. Properties of switching jump diffusions: maximum principles and Harnack inequalities. Bernoulli, 2019, 25: 1045–1075 https://doi.org/10.3150/17-BEJ1012
3	X Chen, Z Chen, K Tran, G Yin. Recurrence and ergodicity for a class of regime-switching jump diffusions. Appl Math Optim, 2019, 80: 415–445 https://doi.org/10.1007/s00245-017-9470-9
4	X X Guo, W Sun. Periodic solutions of stochastic differential equations driven by Lévy noises. J Nonlinear Sci, 2021, 31: 32 https://doi.org/10.1007/s00332-021-09686-5
5	H Hu, L Xu. Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations. J Math Anal Appl, 2018, 466: 896–926 https://doi.org/10.1016/j.jmaa.2018.06.025
6	R Z Khasminskii. Stochastic Stability of Differential Equations. 2nd ed. Berlin: Springer-Verlag, 2012 https://doi.org/10.1007/978-3-642-23280-0
7	E N Lorenz. Deterministic nonperiodic ow. J Atmos Sci, 1963, 20: 130–141 https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
8	X Mao, C Yuan. Stochastic Differential Equations with Markovian Switching. London: Imperial College Press, 2006 https://doi.org/10.1142/p473
9	S P Meyn, R L Tweedie. Markov Chains and Stochastic Stability. Comm Control Engrg Ser. London: Springer-Verlag, 1993 https://doi.org/10.1007/978-1-4471-3267-7
10	D H Nguyen, G Yin. Modeling and analysis of switching diffusion systems: past-dependent switching with a countable state space. SIAM J Control Optim, 2016, 54: 2450–2477 https://doi.org/10.1137/16M1059357
11	D H Nguyen, G Yin. Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space. Potential Anal, 2018, 48: 405–435 https://doi.org/10.1007/s11118-017-9641-y
12	J Shao. Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space. SIAM J Control Optim, 2015, 4: 2462–2479 https://doi.org/10.1137/15M1013584
13	F Xi. Asymptotic properties of jump-diffusion processes with state-dependent switching. Stochastic Process Appl, 2009, 119: 2198–2221 https://doi.org/10.1016/j.spa.2008.11.001
14	F Xi, G Yin. Jump-diffusions with state-dependent switching: existence and uniqueness, Feller property, linearization, and uniform ergodicity. Sci China Math, 2011, 12: 2651–2667 https://doi.org/10.1007/s11425-011-4281-y
15	F Xi, G Yin, C Zhu. 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 https://doi.org/10.1007/978-3-030-25498-8_23
16	F Xi, C Zhu. 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 https://doi.org/10.1137/16M1087837
17	G Yin, C Zhu. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability, Vol 63.New York: Springer, 2010 https://doi.org/10.1007/978-1-4419-1105-6
18	X Zhang, K Wang, D Li. Stochastic periodic solutions of stochastic differential equations driven by Lévy process. J Math Anal Appl, 2015, 430: 231–242 https://doi.org/10.1016/j.jmaa.2015.04.090

[1]	Xiaofei ZHANG, Chungen LIU. Minimal period estimates on P-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems[J]. Front. Math. China, 2021, 16(1): 239-253.
[2]	Yanmin NIU, Xiong LI. Dynamical behaviors for generalized pendulum type equations with p-Laplacian[J]. Front. Math. China, 2020, 15(5): 959-984.
[3]	Xingfan CHEN, Fei GUO, Peng LIU. Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions[J]. Front. Math. China, 2018, 13(6): 1313-1323.
[4]	Xiaoyou LIU. Existence of anti-periodic solutions for hemivariational inequalities[J]. Front. Math. China, 2018, 13(3): 607-618.
[5]	Xindong XU. Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential[J]. Front. Math. China, 2018, 13(1): 227-254.
[6]	Yiju WANG,Kaili ZHANG,Hongchun SUN. Criteria for strong H-tensors[J]. Front. Math. China, 2016, 11(3): 577-592.
[7]	Naihuan JING,Chunhua WANG. Modules for double affine Lie algebras[J]. Front. Math. China, 2016, 11(1): 89-108.
[8]	Chen LING, Liqun QI. l^k,s-Singular values and spectral radius of rectangular tensors[J]. Front Math Chin, 2013, 8(1): 63-83.
[9]	Duokui YAN. A simple existence proof of Schubart periodic orbit with arbitrary masses[J]. Front Math Chin, 2012, 7(1): 145-160.
[10]	Meirong ZHANG. From ODE to DDE[J]. Front Math Chin, 2009, 4(3): 585-598.
[11]	Jiansheng GENG. Almost periodic solutions for a class of higher dimensional Schr?dinger equations[J]. Front Math Chin, 2009, 4(3): 463-482.
[12]	Qiuxiang FENG, Rong YUAN. Existence of almost periodic solutions for neutral delay difference systems[J]. Front Math Chin, 2009, 4(3): 437-462.
[13]	BERTI Massimiliano, BOLLE Philippe. Cantor families of periodic solutions for completely resonant wave equations[J]. Front. Math. China, 2008, 3(2): 151-165.

Viewed

Full text

Abstract

Cited

Shared

Discussed