Strong convergence rate for slow-fast stochastic differential equations with Markovian switching

Yuhan Liao; Kexin Liu; Xiaobin Sun; Liqiong Wang

doi:10.3934/dcdsb.2023011

Article Contents

2023, Volume 28, Issue 8: 4281-4292. Doi: 10.3934/dcdsb.2023011

This issue Previous Article On the delayed vector-bias malaria model in an almost periodic environment Next Article Partial regular solution to the coupled spin polarized system in three dimensions

Strong convergence rate for slow-fast stochastic differential equations with Markovian switching

1.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China
2.
Research Institute of Mathematical Science, Jiangsu Normal University, Xuzhou, 221116, China

^*Corresponding author: Xiaobin Sun
^*Corresponding author: Xiaobin Sun

Received: August 2022

Revised: December 2022

Early access: February 2023

Published: August 2023

This work is supported by the National Natural Science Foundation of China (No. 12271219, 11931004, 12090011), Graduate Research and Innovation Program in Jiangsu Province (No. KYCX22_2836), the QingLan Project of Jiangsu Province and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we study the averaging principle for a class of slow-fast stochastic differential equations with Markovian switching, where the slow component is the solution of a stochastic differential equation and the fast component is a Markov chain. Using the technique of Poisson equation and mollifying approximation, the optimal strong convergence order $ 1/2 $ is obtained under the Lipschitz continuous condition.

Keywords:

Mathematics Subject Classification: Primary: 60H10; Secondary: 60J27.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, second Edition, Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781.
[2]	J. Bao, G. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: Averaging principles, Bernoulli, 23 (2017), 645-669. doi: 10.3150/14-BEJ677.
[3]	G. Basak, A. Bisi and M. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202 (1996), 604-622. doi: 10.1006/jmaa.1996.0336.
[4]	C. E. Bréhier, Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component, Stochastic Process. Appl., 130 (2020), 3325-3368. doi: 10.1016/j.spa.2019.09.015.
[5]	S. Cerrai and M. Freidlin, Averaging principle for stochastic reaction-diffusion equations, Probab.Theory Related Fields, 144 (2009), 137-177. doi: 10.1007/s00440-008-0144-z.
[6]	W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm.Pure Appl.Math, 58 (2005), 1544-1585. doi: 10.1002/cpa.20088.
[7]	D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM J. Multiscale Model. Simul., 6 (2007), 577-594. doi: 10.1137/060673345.
[8]	M. Hairer and X.-M. Li, Averaging dynamics driven by fractional Brownian motion, Ann. Probab., 48 (2020), 1826-1860. doi: 10.1214/19-AOP1408.
[9]	Y. Hu, D. Nualart, X. 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.
[10]	R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernetica, 4 (1968), 260-279.
[11]	D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systemss, Commun. Math. Sci., 8 (2010), 999-1020. doi: 10.4310/CMS.2010.v8.n4.a11.
[12]	W. Liu, M. Röckner, X. Sun and Y. Xie, Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients, J. Differential Equations, 268 (2020), 2910-2948. doi: 10.1016/j.jde.2019.09.047.
[13]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, ingapore, 2006. doi: 10.1142/p473.
[14]	Y. Mao and J. Shao, Averaging principle for two time-scale regime-switching processes, preprint, 2022, arXiv: 2212.05218.
[15]	E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation, Ann. Prob., 29 (2001), 1061-1085. doi: 10.1214/aop/1015345596.
[16]	G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, volume 53 of Texts in Applied Mathematics. Springer, New York, 2008.
[17]	B. Pei, Y. Xu and G. Yin, Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes, Stoch. Dyn., 18 (2018), 1850023, 19 pp. doi: 10.1142/S0219493718500235.
[18]	B. Pei, Y. Xu, G. Yin and X. Zhang, Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes, Nonlinear Anal. Hybrid Syst., 27 (2018), 107-124. doi: 10.1016/j.nahs.2017.08.008.
[19]	M. Röckner, X. Sun and Y. Xie, Strong convergence order for slow-fast McKean-Vlasov stochastic differential equations, Ann. Inst. Henri Poincaré Probab. Stat., 57 (2021), 547-576. doi: 10.1214/20-aihp1087.
[20]	M. Röckner and L. Xie, Diffusion approximation for fully coupled stochastic differential equations, Ann. Probab., 49 (2021), 1205-1236. doi: 10.1214/20-aop1475.
[21]	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.
[22]	X. Sun, L. Xie and Y. Xie, Strong and weak convergence rates for slow-fast stochastic differential equations driven by $\alpha$-stable process, Bernoulli, 28 (2022), 343-369. doi: 10.3150/21-bej1345.
[23]	F. Xi, C. Zhu and F. Wu, On strong Feller property, exponential ergodicity and large deviations principle for stochastic damping Hamiltonian systems with state-dependent switching, J. Differential Equations, 286 (2021), 856-891. doi: 10.1016/j.jde.2021.03.041.
[24]	G. Yin and H. Yang, Two-time-scale jump-diffusion models with Markovian switching regimes, Stoch. Stoch. Rep., 76 (2004), 77-99. doi: 10.1080/10451120410001696261.
[25]	G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Applications of Mathematics (New York) 37. New York: Springer, 1998. doi: 10.1007/978-1-4612-0627-9.
[26]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stochastic Modelling and Applied Probability 63. New York: Springer, 2010. doi: 10.1007/978-1-4419-1105-6.
[27]	C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 103 (2003), 277-291. doi: 10.1016/S0304-4149(02)00230-2.