In this paper, we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays. Under certain assumptions, we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability, respectively. The convergence order is also estimated in terms of noise intensity. Finally, an example with numerical simulation is given to illustrate the theoretical result.
Guangjun SHEN
,
Wentao XU
,
Jiang-Lun WU
. AN AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS DRIVEN BY TIME-CHANGED LÉVY NOISE[J]. Acta mathematica scientia, Series B, 2022
, 42(2)
: 540
-550
.
DOI: 10.1007/s10473-022-0208-7
[1] Applebaum D. Lévy Processes and Stochastic Calculus. Vol 116 of Cambridge Studies in Advanced Mathematics. Cambridge:Cambridge University Press, 2009
[2] Chekroun M D, Simonnet E, Ghil M. Stochastic climate dynamics:random attractors and time-dependent invariant measures. Physica D, 2011, 240(21):1685-1700
[3] Deng C, Liu W. Semi-implicit Euler-Maruyama method for non-linear time-changed stochastic differential equations. BIT Numer Math, 2020, 60(4):1133-1151
[4] Dong Z, Sun X, Xiao H, Zhai J. Averaging principle for one dimensional stochastic Burgers equation. J Differential Equations, 2018, 265(10):4749-4797
[5] Hahn M, Kobayashi K, Ryvkina J, Umarov S. On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations. Electron Comm Probab, 2011, 16:150-164
[6] Jin S, Kobayashi K. Strong approximation of stochastic differential equations driven by a time-changed Brownian motion with time-space-dependent coefficients. J Math Anal Appl, 2019, 476(2):619-636
[7] Khasminskii R. On the principle of averaging the Itô stochastic differential equations. Kibernetika, 1968, 4:260-279
[8] Kobayashi K. Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J Theoret Probab, 2011, 24(3):789-820
[9] Liu W, Mao X, Tang J, Wu Y. Truncated Euler-Maruyama method for classical and time-changed nonautonomous stochastic differential equations. Appl Numer Math, 2020, 153:66-81
[10] Luo D, Zhu Q, Luo Z. An averaging principle for stochastic fractional differential equations with time-delays. Appl Math Lett, 2020, 105:106290
[11] Mao X. Approximate solutions for stochastic differential equations with pathwise uniqueness. Stoch Anal Appl, 1994, 12(3):355-367
[12] Mijena J, Nane E. Space-time fractional stochastic partial differential equations. Stochastic Process Appl, 2015, 125(9):3301-3326
[13] Nane E, Ni Y. Stochastic solution of fractional Fokker-Planck equations with space-time-dependent coefficients. J Math Anal Appl, 2016, 442(1):103-116
[14] Nane E, Ni Y. Stability of the solution of stochastic differential equation driven by time-changed Lévy noise. Proc Amer Math Soc, 2017, 145(7):3085-3104
[15] Nane E, Ni Y. Path stability of stochastic differential equations driven by time-changed Lévy noises. ALEA Lat Am J Probab Math Stat, 2018, 15(1):479-507
[16] Shen G, Wu J-L, Yin X. Averaging principle for fractional heat equations driven by stochastic measures. Appl Math Lett, 2020, 106:106404
[17] Shen G, Song J, Wu J-L. Stochastic averaging principle for distribution dependent stochastic differential equations. Appl Math Lett, 2021. doi:https://doi.org/10.1016/j.aml.2021.107761
[18] Umarov S, Hahn M, Kobayashi K. Beyond the Triangle:Brownian Motion, Itô Calculus, and Fokker-Planck Equation-Fractional Generalizations. Singapore:World Scientific Publishing, 2018
[19] Wu F, Yin G. An averaging principle for two-time-scale stochastic functional differential equations. J Differential Equations, 2020, 269(1):1037-1077
[20] Wu Q. Stability analysis for a class of nonlinear time changed systems. Cogent Mathematics, 2016, 3:1228273
[21] Xu Y, Duan J, Xu W. An averaging principle for stochastic dynamical systems with Lévy noise. Phys D, 2011, 240(17):1395-1401
