zbMATH — the first resource for mathematics

Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes. (English) Zbl 1394.60036

60G22 Fractional processes, including fractional Brownian motion
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Alòs, E.; Mazet, O.; Nualart, D., Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29, 766-801, (1999) · Zbl 1015.60047
[2] J. Bao, Q. Song, G. Yin and C. Yuan, Ergodicity and strong limit results for two-time scale functional stochastic differential equations, to appear in Stoch. Anal. Appl. · Zbl 1388.60076
[3] Bao, J.; Yin, G.; Yuan, C., Two-time scale stochastic partial differential equations driven by alpha-stable noise: averaging principles, Bernoulli, 23, 645-669, (2017) · Zbl 1360.60118
[4] Bao, J.; Yuan, C., Numerical analysis for neutral SPDEs driven by \(\alpha\)-stable processes, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17, 1450031, (2014) · Zbl 1322.65013
[5] Bezandry, P., Existence of almost periodic solutions for semilinear stochastic evolution equations driven by fractional Brownian motion, Electronic J. Differential Equations, 156, 1-21, (2012) · Zbl 1260.60115
[6] Caraballo, T.; Garrido-Atienza, M.; Taniguchi, T., The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal. Theory, Methods Appl., 74, 3671-3684, (2011) · Zbl 1218.60053
[7] Cerrai, S., A khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19, 899-948, (2009) · Zbl 1191.60076
[8] Cerrai, S.; Freidlin, M., Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Th. Relat. Fields, 144, 137-177, (2009) · Zbl 1176.60049
[9] Decreusefond, L.; Üstünel, A., Fractional Brownian motion: theory and applications, ESAIM: Proc., 5, 75-86, (1998) · Zbl 0914.60019
[10] Duan, J.; Wang, W., Effective Dynamics of Stochastic Partial Differential Equations, (2014), Elsevier · Zbl 1298.60006
[11] Fan, X.; Yuan, C., Lyapunov exponents of PDEs driven by fractional noise with Markov switching, Statist. Probab. Lett., 110, 39-50, (2016) · Zbl 1336.60119
[12] Freidlin, M.; Wentzell, A., Random Perturbations of Dynamical Systems, (2012), Springer
[13] Fu, H.; Wan, L.; Liu, J., Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stoch. Process. Appl., 125, 3255-3279, (2015) · Zbl 1322.60111
[14] Fu, H.; Wan, L.; Wang, Y.; Liu, J., Strong convergence rate in averaging principle for stochastic Fitzhugh-Nagumo system with two time-scales, J. Math. Anal. Appl., 416, 609-628, (2014) · Zbl 1325.60107
[15] Hurst, H., Long-term storage capacity of reservoirs, Trans. Amer. Soc. Civil Eng., 116, 770-808, (1951)
[16] Khasminskii, R., A limit theorem for the solutions of differential equations with random right-hand sides, Th. Probab. Appl., 11, 390-406, (1966)
[17] Khasminskii, R.; Yin, G., Limit behavior of two-time scale diffusions revisited, J. Differential Equations, 212, 85-113, (2005) · Zbl 1112.35014
[18] Kolmogorov, A., Wienersche spiralen und einige andere interessante kurven im hilbertschen raum, CR (Dokl.) Acad. Sci. URSS, 26, 115-118, (1940) · JFM 66.0552.03
[19] Mandelbrot, B.; VanNess, J., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437, (1968) · Zbl 0179.47801
[20] Mao, X.; Yuan, C., Stochastic Differential Equations with Markovian Switching, (2006), Imperical College Press · Zbl 1126.60002
[21] Maslowski, B.; Nualart, D., Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202, 277-305, (2003) · Zbl 1027.60060
[22] Mishura, Y., Stochastic Calculus for Fractional Brownian Motion and Related Processes, (2008), Springer · Zbl 1138.60006
[23] Nualart, D.; Rǎscanu, A., Differential equations driven by fractional Brownian motion, Collect. Math., 53, 55-81, (2002) · Zbl 1018.60057
[24] Pei, B.; Xu, Y.; Wu, J. L., Two-time scales hyperbolic parabolic equations driven by Poisson random measures: existence, uniqueness and averaging principles, J. Math. Anal. Appl., 447, 243-268, (2017) · Zbl 1387.60102
[25] B. Pei, Y. Xu, G. Yin and X. Zhang, Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time scale Markovian switching processes, preprint, 2016. · Zbl 1380.60060
[26] Tindel, S.; Tudor, C.; Viens, F., Stochastic evolution equations with fractional Brownian motion, Probab. Th. Relat. Fields, 127, 186-204, (2003) · Zbl 1036.60056
[27] Wu, F.; Yin, G.; Wang, L., Moment exponential stability of random delay systems with two-time scale Markovian switching, Nonlinear Anal. Real World Appl., 13, 2476-2490, (2012) · Zbl 1263.60062
[28] Xu, Y.; Duan, J.; Xu, W., An averaging principle for stochastic dynamical systems with Lévy noise, Physica D: Nonlinear Phenomena, 240, 1395-1401, (2011) · Zbl 1236.60060
[29] Xu, Y.; Guo, R.; Liu, D.; Zhang, H.; Duan, J., Stochastic averaging principle for dynamical systems with fractional Brownian motion, Disc. Contin. Dynam. Syst. Ser. B, 19, 1197-1212, (2014) · Zbl 1314.60122
[30] Xu, Y.; Pei, B.; Guo, R., Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Disc. Contin. Dynam. Syst. Ser. B, 20, 2257-2267, (2015) · Zbl 1335.34090
[31] Xu, Y.; Pei, B.; Li, Y., Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 38, 2120-2131, (2015) · Zbl 1345.60051
[32] Xu, Y.; Pei, B.; Wu, J., Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dynam., 17, 1750013, (2017) · Zbl 1365.34102
[33] Yin, G.; Kan, S.; Wang, L. Y.; Xu, C., Identification of systems with regime switching and unmodeled dynamics, IEEE Trans. Automatic Control, 54, 34-47, (2009) · Zbl 1367.93685
[34] Yin, G.; Zhang, Q., Continuous-Time Markov Chains and Applications: A Two-Time Scale Approach, (2013), Springer · Zbl 1277.60127
[35] Yin, G.; Zhu, C., Hybrid Switching Diffusions: Properties and Applications, (2010), Springer · Zbl 1279.60007
[36] Yin, G., Asymptotic expansions of option price under regime-switching diffusions with a fast varying switching process, Asympt. Anal., 65, 203-222, (2009) · Zbl 1186.60056
[37] Zhang, Q., Hybrid filtering for linear systems with non-Gaussian disturbances, IEEE Trans. Automatic Control, 45, 50-61, (2000) · Zbl 0978.93075
[38] Zhou, X. Y.; Yin, G., Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42, 1466-1482, (2003) · Zbl 1175.91169
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.