Averaging principle for two-time-scale MSDEs driven by Poisson processes

doi:10.3969/j.issn.1000-1565.2025.04.001

Abstract

Abstract: This paper focuses on a class of multi-valued stochastic differential equations(MSDEs)driven by Poisson processes. By applying Itôs formula and the averaging principle, the corresponding averaged system is derived, and a quantitative comparison of their solutions is conducted. The results show that, under certain assumptions, the solution of the original system converges to that of the averaged system in the mean-square sense. This conclusion not only expands the application scope of the averaging principle in stochastic systems but also provides a new theoretical tool for the approximate analysis of multi-scale stochastic dynamical systems.

Key words: MSDEs, Poisson processes, stochastic averaging principle

CLC Number:

O211.63

XU Yan,GAO Peixuan. Averaging principle for two-time-scale MSDEs driven by Poisson processes[J]. Journal of Hebei University(Natural Science Edition), 2025, 45(4): 337-342.

References

[1] DUAN J Q. An introduction to stochastic dynamics[M]. New York:Cambridge University Press, 2015.
[2] 李志阐.随机微分方程解的性质[J].河北大学学报(自然科学版), 1964: 23-30.
[3] 赵桂华,王黎明.带泊松跳马尔可夫调制随机微分方程的渐近稳定性[J].河北大学学报(自然科学版), 2011, 31(6): 578-580.
[4] KHASMINSKII R Z. On the principle of averaging the Itô stochastic differential equations[J]. Kybernetika, 1968,4(3): 260-279.
[5] KOLOMIETS V G, MELNIKOV A I. Averaging of stochastic systems of integral-differential equations with Poisson noise[J]. Ukr Math J, 1991, 43(2): 242-246. DOI: 10.1007/BF01060515.
[6] XU Y, DUAN J Q, XU W. An averaging principle for stochastic dynamical systems with Lévy noise[J]. Phys D Nonlinear Phenom, 2011, 240(17): 1395-1401. DOI: 10.1016/j.physd.2011.06.001.
[7] XU Y, PEI B, LI Y G. Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise[J]. Math Meth Appl Sci, 2015, 38(11): 2120-2131. DOI: 10.1002/mma.3208.
[8] XU Y, GUO R, LIU D, et al. Stochastic averaging principle for dynamical systems with fractional Brownian motion[J]. Discrete Contin Dyn Syst B, 2014, 19(4): 1197-1212. DOI: 10.3934/dcdsb.2014.19.1197.
[9] XU Y, PEI B, GUO R. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion[J]. Discrete Contin Dyn Syst B, 2015, 20(7): 2257-2267. DOI: 10.3934/dcdsb.2015.20.2257.
[10] XU Y, PEI B, WU J L. Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion[J]. Stoch Dyn, 2017, 17(2): 1750013. DOI: 10.1142/s0219493717500137.
[11] STRAWDERMAN R, YIN G, ZHANG Q. Continuous-time Markov chains and applications: a singular perturbation approach[J]. J Am Stat Assoc, 1998, 93(444): 1529. DOI: 10.2307/2670080.
[12] WU F K, 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.
[13] YIN G, TALAFHA Y, XI F B. Stochastic Liénard equations with random switching and two-time scales[J]. Commun Stat Theory Meth, 2014, 43(7): 1533-1547. DOI: 10.1080/03610926.2012.741741.
[14] YIN G, YANG H. Two-time-scale jump-diffusion models with Markovian switching regimes[J]. Stoch Stoch Rep, 2004, 76(2): 77-99. DOI: 10.1080/10451120410001696261.
[15] XU J, LIU J C, YU M. Strong averaging principle for two-time-scale SDEs with nonLipschitz coefficients[J]. J Math Anal Appl, 2018, 468(1): 116-140.
[16] LIU S T. Mixed slow-fast stochastic differential equations: Averaging principle result[J]. Fract Calc Appl Anal, 2025, 28(1): 181-207. DOI: 10.1007/s13540-024-00368-z.
[17] 王培光, 候娟娟. 平均法在不确定系统稳定性分析中的应用[J]. 河北大学学报(自然科学版), 2017, 37(1): 1-4. DOI: 10.3969/j.issn.1000-1565.2017.01.001.
[18] SUN X B, XIA H L, XIE Y C, et al. Strong averaging principle for a class of slow-fast singular SPDEs driven by α-stable process[J]. Front Math, 2023, 18(3): 565-590. DOI: 10.1007/s11464-021-0069-8.
[19] 徐燕, 李彩月. 具有随机切换的泛函Liénard方程的平均法[J]. 河北大学学报(自然科学版), 2023, 43(1): 9-15. DOI: 10.3969/j.issn.1000-1565.2023.01.002.
[20] ZHANG Y J, HUANG Q, WANG X, et al. Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes[J]. J Differ Equ, 2024, 379: 721-761. DOI: 10.1016/j.jde.2023.10.031.
[21] CÉPA E. Problàme de skorohod multivoque[J]. Ann Probab, 1998, 26(2): 500-532. DOI: 10.1214/aop/1022855642
[22] XU J, LIU J C. An averaging principle for multivalued stochastic differential equations[J]. Stoch Anal Appl, 2014, 32(6): 962-974. DOI: 10.1080/07362994.2014.959594.
[23] GUO R, PEI B. Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson point Processes[J]. Stoch Anal Appl, 2018, 36(4): 751-766. DOI: 10.1080/07362994.2018.1461567.
[24] ZĂLINESCU A. Weak solutions and optimal control for multivalued stochastic differential equations[J]. Nonlinear Differ Equ Appl Nodea, 2008, 15(4): 511-533. DOI: 10.1007/s00030-008-7037-9. (