Stochastic Averaging Principle for Two-Time-Scale SDEs with Distribution-Dependent Coefficients Driven by Fractional Brownian Motion

Abstract

In this paper, we derive an averaging principle for a fast–slow system of stochastic differential equations (SDEs) involving distribution-dependent coefficients driven by both fractional Brownian motion (fBm) and standard Brownian motion (Bm). We first establish the existence and uniqueness of solutions of the fast–slow system and the corresponding averaging equation. Then, we show that the slow component strongly converges to the solution of the associated averaged equation.

Keywords

Averaging principle / Fast–slow systems / Fractional Brownian motion / Standard Brownian motion / 60H10 / 60G22 / 34C29 / 35Q83

References

[1] Alòs E, Nualart D. Stochastic integration with respect to the fractional Brownian motion. Stochastics, 2003, 75: 129-152

[2] Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic calculus for fractional Brownian motion and applications, 2008, London, Springer-Verlag

[3] Cao G, He K. On a type of stochastic differential equations driven by countably many Brownian motions. J. Funct. Anal., 2003, 203: 262-285

[4] Dong Z, Sun X, Xiao H, Zhai J. Averaging principle for one dimensional stochastic Burgers equation. J. Differ. Equ., 2018, 265: 4749-4797

[5] Fan X, Huang X, Suo Y, Yuan C. Distribution dependent SDEs driven by fractional Brownian motions. Stoch. Process. Appl., 2022, 151: 23-67

[6] Fouque, J. P., Papanicolaou, G., Sircar, R., Solna, K.: Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press (2011)

[7] Givon D. Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems. Multisc. Model. Simul., 2007, 6: 577-594

[8] Guo Z, Lv G, Wei J. Averaging principle for stochastic differential equations under a weak condition. Chaos, 2020, 30 123139

[9] Hairer M, Li X-M. Averaging dynamics driven by fractional Brownian motion. Ann. Probab., 2020, 48: 1826-1860

[10] Hu Y. Analysis on Gaussian spaces, 2017, Hackensack, World Scientific Publishing Co. Pte. Ltd.

[11] Huang X, Wang F-Y. Distribution dependent SDEs with singular coefficients. Stoch. Process. Appl., 2019, 129: 4747-4770

[12] Hong W, Li S, Liu W. Strong convergence rates in averaging principle for slow-fast McKean–Vlasov SPDEs. J. Differ. Equ., 2022, 316: 94-135

[13] Huang X, Ren P, Wang F-Y. Distribution dependent stochastic differential equations. Front. Math. China., 2021, 16: 257-301

[14] Kac, M.: Foudations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 1954-1955. vol. III. pp. 171–197. University of California Press, Berkeley and Los Angeles (1956)

[15] Kac M. Probability and related topics in the physical sciences, 1959, New York, Interscience Publishers

[16] Khasminskii R. On the principle of averaging the Itô stochastic differential equations. Kybernetika, 1968, 4: 260-279

[17] Li Y, Mao X, Song Q, Wu F, Yin G. Strong convergence of Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations under local Lipschitz conditions of state variables. IMA J. Numer. Anal., 2022

[18] Liu W, Röckner M, Sun X, Xie Y. Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients. J. Differ. Equ., 2020, 268: 2910-2948

[19] Luo D, Zhu Q, Luo Z. An averaging principle for stochastic fractional differential equations with time-delays. Appl. Math. Lett., 2020, 105 106290

[20] McKean HP. A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA, 1966, 56: 1907-1911

[21] Mehri S, Stannat W. Weak solutions to Vlasov–McKean equations under Lyapunov-type conditions. Stoch. Dyn., 2019, 19: 1950042

[22] Mishura, Y.: Stochastic calculus for fractional Brownian motion and related processes. Lect. Notes Math. 1929 (2008)

[23] Nualart D. Malliavin Calculus and Related Topics, 20062New York, Springer

[24] Pei B, Xu Y, Wu J-L. Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Appl. Math. Lett., 2020, 100 106006

[25] Pei B, Inahama Y, Xu Y. Averaging principle for fast-slow system driven by mixed fractional Brownian rough path. J. Differ. Equ., 2021, 301: 202-235

[26] Pei, B., Inahama, Y., Xu, Y.: Averaging principles for mixed fast-slow systems driven by fractional Brownian. arXiv:2001.06945v4

[27] Röckner M, Sun X, Xie Y. Strong convergence order for slow-fast McKean–Vlasov stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat., 2021, 57: 547-576

[28] Shen G, Song J, Wu J-L. Stochastic averaging principle for distribution dependent stochastic differential equations. Appl. Math. Lett., 2022, 125 107761

[29] Shen G, Xiang J, Wu J-L. Averaging principle for distribution dependent stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. J. Differ. Equ., 2022, 321: 381-414

[30] Shen G, Wu J-L, Xiao R, Yin X. An averaging principle for neutral stochastic fractional order differential equations with variable delays driven by Lévy noise. Stoch. Dyn., 2022, 22: 2250009

[31] Wang F-Y. Distribution dependent SDEs for Landau type equations. Stoch. Process. Appl., 2018, 128: 595-621

[32] Wu F, Tian T, Rawlings JB, Yin G. Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations. J. Chem. Phys., 2016, 144 174112

[33] Xi F, Zhu C. Jump type stochastic differential equations with non-lipschitz coefficients: non confluence, Feller and strong Feller properties, and exponential ergodicity. J. Differ. Equ., 2019, 266: 4668-4711

[34] Xu J, Liu J, Miao Y. Strong averaging principle for two-time-scale SDEs with non-Lipschitz coefficients. J. Math. Anal. Appl., 2018, 468: 116-140

[35] Xu, J., Liu, J.: Stochastic averaging principle for two time-scale jump-diffusion SDEs under the non-Lipschitz coefficients. Stochastics (2020). https://doi.org/10.1080/17442508.2020.1784897

[36] Xu J, Liu J, Liu J, Miao Y. Strong averaging principle for two-time-scale stochastic McKean–Vlasov equations. Appl. Math. Optim., 2021, 84(Suppl 1): 837-867

[37] Xu Y, Duan J, Xu W. An averaging principle for stochastic dynamical systems with Lévy noise. Phys. D, 2011, 240: 1395-1401

[38] Xu Y, Pei B, Wu J-L. Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. Stoch. Dyn., 2017, 17: 1750013

Funding

National Natural Science Foundation of China(12071003)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature