Averaging principle for two-time-scale stochastic functional differential equations with past-dependent switching

Minyu Wu; Xizhong Yang; Feiran Yuan; Xuyi Qiu; Minyu Wu; Xizhong Yang; Feiran Yuan; Xuyi Qiu

doi:10.3934/math.2025017

AIMS Mathematics

2025, Volume 10, Issue 1: 353-387. doi: 10.3934/math.2025017

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Averaging principle for two-time-scale stochastic functional differential equations with past-dependent switching

National Key Laboratory of Avionics Integration and Aviation System-of-Systems Synthesis, Shanghai 201100, China

Received: 18 October 2024 Revised: 26 December 2024 Accepted: 28 December 2024 Published: 08 January 2025
MSC : 34K26, 34K40, 60H10

This paper aimed to establish the averaging principle for the two-timescale stochastic functional differential equations with past-dependent switching. Initially, the existence and uniqueness of solutions, as well as moment estimates, were obtained by using classical interlacing techniques. Furthermore, the interaction between the fast and slow processes was derived based on the properties of the Poisson random measure. Subsequently, employing the coupling method and the integration by parts formula for the generator, the exponential ergodicity of the frozen Markov chain and the Lipschitz continuity of its invariant measures were proved. In addition to the challenges posed by the dependence on history, the Lipschitz condition under uniform norms that the generator satisfies also introduced computational and proof difficulties. Therefore, more refined estimates were provided for the segment process. Based on these results, together with weak convergence and martingale methods, the averaging principle for the original system was established. Finally, two examples were provided to illustrate the differences between the results presented here and the classical results.
- stochastic functional differential equation,
- past-dependent switching,
- averaging principle,
- coupling method,
- weak convergence,
- martingale method
Citation: Minyu Wu, Xizhong Yang, Feiran Yuan, Xuyi Qiu. Averaging principle for two-time-scale stochastic functional differential equations with past-dependent switching[J]. AIMS Mathematics, 2025, 10(1): 353-387. doi: 10.3934/math.2025017

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Abstract

This paper aimed to establish the averaging principle for the two-timescale stochastic functional differential equations with past-dependent switching. Initially, the existence and uniqueness of solutions, as well as moment estimates, were obtained by using classical interlacing techniques. Furthermore, the interaction between the fast and slow processes was derived based on the properties of the Poisson random measure. Subsequently, employing the coupling method and the integration by parts formula for the generator, the exponential ergodicity of the frozen Markov chain and the Lipschitz continuity of its invariant measures were proved. In addition to the challenges posed by the dependence on history, the Lipschitz condition under uniform norms that the generator satisfies also introduced computational and proof difficulties. Therefore, more refined estimates were provided for the segment process. Based on these results, together with weak convergence and martingale methods, the averaging principle for the original system was established. Finally, two examples were provided to illustrate the differences between the results presented here and the classical results.

References

[1]	Y. Li, F. Wu, G. Yin, Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4417–4443. http://doi.org/10.3934/dcdsb.2019125 doi: 10.3934/dcdsb.2019125
[2]	F. Wu, T. Tian, J. B. Rawlings, Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations, J. Chem. Phys., 144 (2016), 174112. http://doi.org/10.1063/1.4948407 doi: 10.1063/1.4948407
[3]	L. Hu, X. Mao, Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control Lett., 62 (2013), 178–187. http://doi.org/10.1016/j.sysconle.2012.11.009 doi: 10.1016/j.sysconle.2012.11.009
[4]	H. Qian, Z. Wen, G. Yin, Numerical solutions for optimal control of stochastic Kolmogorov systems with regime-switching and random jumps, Stat. Infer. Stoch. Process., 25 (2022), 105–125. http://doi.org/10.1007/s11203-021-09267-z doi: 10.1007/s11203-021-09267-z
[5]	G. Yin, Q. Zhang, Continuous-time Markov chains and applications: a two-time-scale approach, Springer, 2013. http://doi.org/10.1007/978-1-4614-4346-9
[6]	R. Wang, X. Li, G. Yin, Asymptotic properties of multi-species Lotka-Volterra models with regime switching involving weak and strong interactions, J. Nonlinear Sci., 30 (2020), 565–601. http://doi.org/10.1007/s00332-019-09583-y doi: 10.1007/s00332-019-09583-y
[7]	G. Yin, X. Mao, C. Yuan, D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: numerical algorithms and existence and uniqueness of solutions, SIAM J. Math. Anal., 41 (2010), 2335–2352. http://doi.org/10.1137/080727191 doi: 10.1137/080727191
[8]	A. Budhiraja, P. Dupuis, A. Ganguly, Large deviations for small noise diffusions in a fast Markovian environment, Electron. J. Probab., 23 (2018), 112. http://doi.org/10.1214/18-EJP228 doi: 10.1214/18-EJP228
[9]	D. T. Nguyen, G. Yin, Asymptotic expansions of solutions of systems of Kolmogorov backward equations for two-time-scale switching diffusions, Q. Appl. Math., 71 (2013), 601–628. http://doi.org/10.1090/S0033-569X-2013-01277-X doi: 10.1090/S0033-569X-2013-01277-X
[10]	D. H. Nguyen, G. Yin, Recurrence for switching diffusion with past dependent switching and countable state space, Math. Control Relat. Fields, 8 (2018), 879–897. http://doi.org/10.3934/mcrf.2018039 doi: 10.3934/mcrf.2018039
[11]	D. H. Nguyen, G. Yin, Modeling and analysis of switching diffusion systems: past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450–2477. http://doi.org/10.1137/16M1059357 doi: 10.1137/16M1059357
[12]	D. H. Nguyen, G. Yin, Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space, Potential Anal., 48 (2018), 405–435. http://doi.org/10.1007/s11118-017-9641-y doi: 10.1007/s11118-017-9641-y
[13]	R. Z. Khasminskii, G. Yin, On transition densities of singularly perturbed diffusions with fast and slow components, SIAM J. Appl. Math., 56 (1996), 39–51. http://doi.org/10.1137/S0036139995282906 doi: 10.1137/S0036139995282906
[14]	E. Pardoux, A. Y. Veretennikov, On the Poisson equation and diffusion approximation 1, Ann. Probab., 29 (2001), 1061–1085. http://doi.org/10.1214/aop/1015345596 doi: 10.1214/aop/1015345596
[15]	E. Pardoux, A. Y. Veretennikov, On the Poisson equation and diffusion approximation 2, Ann. Probab., 31 (2003), 1166–1192. http://doi.org/10.1214/aop/1055425774 doi: 10.1214/aop/1055425774
[16]	E. Pardoux, A. Y. Veretennikov, On the Poisson equation and diffusion approximation 3, Ann. Probab., 33 (2005), 1111–1133. http://doi.org/10.1214/009117905000000062 doi: 10.1214/009117905000000062
[17]	M. Röckner, L. Xie, Averaging principle and normal deviations for multiscale stochastic systems, Commun. Math. Phys., 383 (2021), 1889–1937. http://doi.org/10.1007/s00220-021-04069-z doi: 10.1007/s00220-021-04069-z
[18]	M. Röckner, L. Xie, L. Yang, Averaging principle and normal deviations for multi-scale stochastic hyperbolic-parabolic equations, Stoch. Partial Differ. Equations, 11 (2023), 869–907. http://doi.org/10.1007/s40072-022-00248-8 doi: 10.1007/s40072-022-00248-8
[19]	R. Z. Khasminskii, On the principle of averaging the Itô's stochastic differential equations, Kybernetika, 4 (1968), 260–279.
[20]	H. J. Kushner, Approximation and weak convergence methods for random processes, with applications to stochastic systems theory, MIT Press, 1985. http://doi.org/10.2307/2288588
[21]	M. Han, B. Pei, An averaging principle for stochastic evolution equations with jumps and random time delays, AIMS Math., 6 (2021), 39–51. http://doi.org/10.3934/math.2021003 doi: 10.3934/math.2021003
[22]	D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999–1020. http://doi.org/10.4310/CMS.2010.v8.n4.a11 doi: 10.4310/CMS.2010.v8.n4.a11
[23]	J. Shao, Strong solutions and strong Feller properties for regime-switching diffusion processes in an inﬁnite state space, SIAM J. Control Optim., 53 (2025), 2462–2479. http://doi.org/10.1137/15M1013584 doi: 10.1137/15M1013584
[24]	J. Shao, K. Zhao, Continuous dependence for stochastic functional differential equations with state-dependent regime-switching on initial values, Acta Math. Sin. Engl. Ser., 37 (2021), 389–407. http://doi.org/10.1007/s10114-020-9205-8 doi: 10.1007/s10114-020-9205-8
[25]	N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, 2 Eds., North-Holland Publishing Company, 1989.
[26]	R. Situ, Theory of stochastic differential equations with jumps and applications, Springer, 2005. http://doi.org/10.1007/b106901
[27]	X. Mao, Stochastic differential equations and applications, 2 Eds., Elsevier, 2008.
[28]	X. Mao, C. Yuan, Stochastic differential equations with Markovian switching, Imperial College Press, 2006. http://doi.org/10.1142/p473
[29]	M. Chen, From Markov chains to non-equilibrium particle systems, 2 Eds., World Scientific, 1992. http://doi.org/10.1142/5513
[30]	I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Springer-Verlag, 1988. http://doi.org/10.1007/978-1-4684-0302-2
[31]	P. Billingsley, Convergence of probability measures, 2 Eds., John Wiley & Sons, Inc., 1999. http://doi.org/10.1002/9780470316962
[32]	A. N. Shiryaev, Probability, 2 Eds., Springer-Verlag, 1996. http://doi.org/10.1007/978-1-4757-2539-1
[33]	F. Wu, G. Yin, An averaging principle for two-time-scale stochastic functional differential equations, J. Differ. Equations, 269 (2020), 1037–1077. http://doi.org/10.1016/j.jde.2019.12.024 doi: 10.1016/j.jde.2019.12.024
[34]	R. Kasinathan, R. Kasinathan, D. Chalishajar, D. Baleanu, V. Sandrasekaran, The averaging principle of Hilfer fractional stochastic pantograph equations with non-Lipschitz conditions, Stat. Probab. Lett., 215 (2024), 110221. http://doi.org/10.1016/j.spl.2024.110221 doi: 10.1016/j.spl.2024.110221

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