Abstract
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.
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References
Applebaum D. Lévy Processes and Stochastic Calculus. Vol 116 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2009
Chekroun M D, Simonnet E, Ghil M. Stochastic climate dynamics: random attractors and time-dependent invariant measures. Physica D, 2011, 240(21): 1685–1700
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
Dong Z, Sun X, Xiao H, Zhai J. Averaging principle for one dimensional stochastic Burgers equation. J Differential Equations, 2018, 265(10): 4749–4797
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
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
Khasminskii R. On the principle of averaging the Itô stochastic differential equations. Kibernetika, 1968, 4: 260–279
Kobayashi K. Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J Theoret Probab, 2011, 24(3): 789–820
Liu W, Mao X, Tang J, Wu Y. Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations. Appl Numer Math, 2020, 153: 66–81
Luo D, Zhu Q, Luo Z. An averaging principle for stochastic fractional differential equations with time-delays. Appl Math Lett, 2020, 105: 106290
Mao X. Approximate solutions for stochastic differential equations with pathwise uniqueness. Stoch Anal Appl, 1994, 12(3): 355–367
Mijena J, Nane E. Space-time fractional stochastic partial differential equations. Stochastic Process Appl, 2015, 125(9): 3301–3326
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
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
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
Shen G, Wu J-L, Yin X. Averaging principle for fractional heat equations driven by stochastic measures. Appl Math Lett, 2020, 106: 106404
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
Umarov S, Hahn M, Kobayashi K. Beyond the Triangle: Brownian Motion, Itô Calculus, and Fokker-Planck Equation-Fractional Generalizations. Singapore: World Scientific Publishing, 2018
Wu F, Yin G. An averaging principle for two-time-scale stochastic functional differential equations. J Differential Equations, 2020, 269(1): 1037–1077
Wu Q. Stability analysis for a class of nonlinear time changed systems. Cogent Mathematics, 2016, 3: 1228273
Xu Y, Duan J, Xu W. An averaging principle for stochastic dynamical systems with Lévy noise. Phys D, 2011, 240(17): 1395–1401
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This research is supported by the National Natural Science Foundation of China (12071003, 11901005) and the Natural Science Foundation of Anhui Province (2008085QA20).
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Shen, G., Xu, W. & Wu, JL. An Averaging Principle for Stochastic Differential Delay Equations Driven by Time-Changed Lévy Noise. Acta Math Sci 42, 540–550 (2022). https://doi.org/10.1007/s10473-022-0208-7
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DOI: https://doi.org/10.1007/s10473-022-0208-7