# American Institute of Mathematical Sciences

• Previous Article
Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation
• DCDS-B Home
• This Issue
• Next Article
Asymptotic pattern of a migratory and nonmonotone population model
June  2014, 19(4): 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

## Stochastic averaging principle for dynamical systems with fractional Brownian motion

 1 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China, China, China, China 2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  April 2013 Revised  September 2013 Published  April 2014

Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$, is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integral of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.
Citation: Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197
##### References:

show all references

##### References:
 [1] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [2] Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 [3] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [4] Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084 [5] David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014 [6] Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 [7] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [8] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [9] Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 [10] Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $1/2$. Evolution Equations & Control Theory, 2021, 10 (4) : 921-935. doi: 10.3934/eect.2020096 [11] Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 [12] Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 [13] Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100 [14] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 [15] Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483 [16] Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 [17] Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287 [18] Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021031 [19] Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247 [20] Kexue Li. Effects of the noise level on nonlinear stochastic fractional heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5437-5460. doi: 10.3934/dcdsb.2019065

2020 Impact Factor: 1.327