Averaging principle for stochastic function Liénard equations with random switching

doi:10.3969/j.issn.1000-1565.2023.01.002

Journal of Hebei University(Natural Science Edition) ›› 2023, Vol. 43 ›› Issue (1): 9-15.DOI: 10.3969/j.issn.1000-1565.2023.01.002

Previous Articles Next Articles

Averaging principle for stochastic function Liénard equations with random switching

XU Yan, LI Caiyue

College of Mathematics and Information Science, Hebei University, Baoding 071002, China

Received:2022-01-19 Online:2023-01-25 Published:2023-02-22

Abstract

Abstract: By using the idea of nearly decomposability and aggregation, functional Itô formula and the averaging principle, this article considers the averaging principle for the stochastic functional Liénard equations with random switching was considered. Because the Markov chain is in a large state space and involves functional diffusions, the original system was dealt with. Under suitable conditions, it shows that the original system converges to a limit process, which is more convenient for calculation and analysis. The properties of the original system are obtained by studying the limiting process.

Key words: random switching, averaging principle, martingale problem formulation, stochastic functional differential equations

CLC Number:

O211.63

XU Yan, LI Caiyue. Averaging principle for stochastic function Liénard equations with random switching[J]. Journal of Hebei University(Natural Science Edition), 2023, 43(1): 9-15.

References

[1] KRYLOV N M, BOGOLYUBOV N N. Les propriétés ergodiques des suites des probabilitiés en chaine [J]. CR Paris, 1937, 204:1454-1456.
[2] GIKHMAN I I. On a theorem of NN Bogoliubov [J]. Ukrain Mat Zh, 1952, 4(2): 215-218.
[3] VOLOSOV V M. Averaging in systems of ordinary differential equations [J]. Russ Math Surv, 1962, 17(6):1-126.DOI: 10.1070/rm1962vo17n06abeh001130.
[4] KHASMINSKII R Z. On the principle of averaging the itô stochastic differential equations [J]. Kybernetika, 1968, 4(3): 260-279.
[5] LIU D. Strong convergence of principle of averaging for multiscale stochastic dynamical system [J]. Commun Math Sci, 2010, 8(4):999-1020. DOI: 10.4310/cms.2010.v8.n4.all.
[6] GIVON D. Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems [J]. Multiscale Modeling & Simul, 2007, 6(2):577-594. DOI: 10.1137/060673345.
[7] LIU D. Strong convergence rate of principle of averaging for jump-diffusion processes [J]. Front Math China, 2012, 7(2):305-320. DOI: 10.1007/s11464-012-0193-6.
[8] WAINRIB G. Double averaging principle for periodically forced stochastic slow-fast systems [J]. Electron Commun Probab, 2013, 18(51):1-12. DOI: 10.1214/ecp.v18-1975.
[9] XU J, MIAO Y, LIU J C. Strong averaging principle for two-time-scale SDEs with non-Lipschitz coefficients [J]. J Math Anal App, 2018, 468(1):116-140. DOI: 10.1016/j.jmaa.2018.07.039.
[10] YIN G, ZHANG Q. Continuous-time Markov chains and applications: A singular perturbation approach [M]. New York: Spring, 1998.
[11] YIN G, YANG H. Two-time-scale jump-diffusion models with Markovian switching regimes [J]. Stoch Rep Group, 2004, 76(2):77-99. DOI: 10.1080/10451120410001696261.
[12] YIN G, TALAFHA Y, XI F. Stochastic liénard equations with random switching and two-time scales [J]. Commun Stat Theory Methods, 2014, 43(7):1533-1547. DOI: 10.1080/03610926.2012.741741.
[13] DUPIRE B. Functional Itô calculus[J]. Quant Finance, 2019, 19(5):721-729.DOI: 10.1080/14697688.2019.1575974.
[14] WU F, YIN G. An averaging principle for two-time-scale stochastic functional differential equations [J]. J Differ Equ, 2020, 269(1):1037-1077. DOI: 10.1016/j.jde.2019.12.024.
[15] BALLY V, CARAMELLINO L, CONT R. Stochastic integration by parts and functional Itô calculus[M]. Switzerland: Springer International Publishing, 2016. (

Averaging principle for stochastic function Liénard equations with random switching

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 0

Recommended Articles

Metrics

Comments