Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
geometric fractional Brownian motion; stochastic differential equations in Hilbert space; stochastic bilinear equation
geometric fractional Brownian motion; stochastic differential equations in Hilbert space; stochastic bilinear equation
Summary:
A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than $1/2$. Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.
References:
[1] Alòs, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 (2002), 129-152. DOI 10.1080/1045112031000078917 | MR 1978896 | Zbl 1028.60048
[2] Bártek, J., Garrido-Atienza, M. J., Maslowski, B.: Stochastic porous media equation driven by fractional Brownian motion. Stoch. Dyn. 13 (2013), Article ID 1350010, 33 pages. DOI 10.1142/S021949371350010X | MR 3116928 | Zbl 1291.35453
[3] Bonaccorsi, S.: Nonlinear stochastic differential equations in infinite dimensions. Stochastic Anal. Appl. 18 (2000), 333-345. DOI 10.1080/07362990008809673 | MR 1758177 | Zbl 0959.60046
[4] Prato, G. Da, Iannelli, M., Tubaro, L.: An existence result for a linear abstract stochastic equation in Hilbert spaces. Rend. Sem. Mat. Univ. Padova 67 (1982), 171-180. MR 0682709 | Zbl 0499.60061
[5] Prato, G. Da, Iannelli, M., Tubaro, L.: Some results on linear stochastic differential equations in Hilbert spaces. Stochastics 6 (1982), 105-116. DOI 10.1080/17442508208833210 | MR 0665246 | Zbl 0475.60041
[6] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44, Cambridge University Press, Cambridge (1992). DOI 10.1017/CBO9780511666223 | MR 1207136 | Zbl 0761.60052
[7] Duncan, T. E., Maslowski, B., Pasik-Duncan, B.: Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stochastic Processes Appl. 115 (2005), 1357-1383. DOI 10.1016/j.spa.2005.03.011 | MR 2152379 | Zbl 1076.60054
[8] Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. Ec. d'Ete Probab. Saint-Flour IV-1974, Lect. Notes Math. 480 1-96 (1975), French. DOI 10.1007/bfb0080190 | MR 0413238 | Zbl 0331.60025
[9] Flandoli, F.: On the semigroup approach to stochastic evolution equations. Stochastic Anal. Appl. 10 (1992), 181-203. DOI 10.1080/07362999208809262 | MR 1154534 | Zbl 0762.60046
[10] Garrido-Atienza, M. J., Maslowski, B., Šnupárková, J.: Semilinear stochastic equations with bilinear fractional noise. Discrete Contin. Dyn. Syst., Ser. B 21 (2016), 3075-3094. DOI 10.3934/dcdsb.2016088 | MR 3567802 | Zbl 1353.60059
[11] León, J. A., Nualart, D.: Stochastic evolution equations with random generators. Ann. Probab. 26 (1998), 149-186. DOI 10.1214/aop/1022855415 | MR 1617045 | Zbl 0939.60066
[12] Mishura, Y. S.: Quasi-linear stochastic differential equations with a fractional-Brownian component. Theory Probab. Math. Statist. 68 (2004), 103-115 English. Ukrainian original translation from Teor. \u Imov\=ır. Mat. Stat. 68 2003 95-106. DOI 10.1090/S0094-9000-04-00608-8 | MR 2000399 | Zbl 1050.60060
[13] Mishura, Y., Shevchenko, G.: The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics 80 (2008), 489-511. DOI 10.1080/17442500802024892 | MR 2456334 | Zbl 1154.60046
[14] Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus. From Stein's Method to Universality. Cambridge Tracts in Mathematics 192, Cambridge University Press, Cambridge (2012). DOI 10.1017/CBO9781139084659 | MR 2962301 | Zbl 1266.60001
[15] Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications, Springer, New York (1995). DOI 10.1007/978-1-4757-2437-0 | MR 1344217 | Zbl 0837.60050
[16] Nualart, D.: Stochastic integration with respect to fractional Brownian motion and applications. Stochastic Models. Seventh Symposium on Probability and Stochastic Processes, Mexico City 2002 Contemp. Math. 336, American Mathematical Society, Providence J. M. Gonzáles-Barrios et al. (2003), 3-39. DOI 10.1090/conm/336 | MR 2037156 | Zbl 1063.60080
[17] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer, New York (1983). DOI 10.1007/978-1-4612-5561-1 | MR 0710486 | Zbl 0516.47023
[18] Pérez-Abreu, V., Tudor, C.: Multiple stochastic fractional integrals: a transfer principle for multiple stochastic fractional integrals. Bol. Soc. Mat. Mex., III. Ser. 8 (2002), 187-203. MR 1952159 | Zbl 1020.60050
[19] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003
[20] Šnupárková, J.: Stochastic bilinear equations with fractional Gaussian noise in Hilbert space. Acta Univ. Carol., Math. Phys. 51 (2010), 49-67. MR 2828153 | Zbl 1229.60073
[21] Tanabe, H.: Equations of Evolution. Monographs and Studies in Mathematics 6, Pitman, London (1979). MR 0533824 | Zbl 0417.35003
Search
Partner of
