| Title:
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Stochastic evolution equations driven by Liouville fractional Brownian motion (English) |
| Author:
|
Brzeźniak, Zdzisław |
| Author:
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van Neerven, Jan |
| Author:
|
Salopek, Donna |
| Language:
|
English |
| Journal:
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Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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62 |
| Issue:
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1 |
| Year:
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2012 |
| Pages:
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1-27 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of ${\cal L}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac 12$ we show that a function $\Phi \colon (0,T)\to {\cal L}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$ {\rm d}U(t) = AU(t) {\rm d}t + B {\rm d}W_H^\beta (t) $$ driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A\colon \scr D(A)\to E$ and $B\colon H\to E$, and the Hurst parameter $\beta $. As an application it is shown that second-order parabolic SPDEs on bounded domains in $\mathbb R^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac {1}{4}d<\beta <1$. (English) |
| Keyword:
|
(Liouville) fractional Brownian motion |
| Keyword:
|
fractional integration |
| Keyword:
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stochastic evolution equations |
| MSC:
|
35R60 |
| MSC:
|
47D06 |
| MSC:
|
60G18 |
| MSC:
|
60H05 |
| idZBL:
|
Zbl 1249.60109 |
| idMR:
|
MR2899731 |
| DOI:
|
10.1007/s10587-012-0011-z |
| . |
| Date available:
|
2012-03-05T07:07:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142035 |
| . |
| Reference:
|
[1] Anh, V. V., Grecksch, W. A.: A fractional stochastic evolution equation driven by fractional Brownian motion.Monte Carlo Methods Appl. 9 (2003), 189-199. Zbl 1049.60056, MR 2009368 |
| Reference:
|
[2] Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes.Ann. Probab. 29 (2001), 766-801. MR 1849177, 10.1214/aop/1008956692 |
| Reference:
|
[3] Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and its Applications.Springer London (2008). MR 2387368 |
| Reference:
|
[4] Brze'zniak, Z.: On stochastic convolutions in Banach spaces and applications.Stochastics Stochastics Rep. 61 (1997), 245-295. MR 1488138, 10.1080/17442509708834122 |
| Reference:
|
[5] Brze'zniak, Z., Neerven, J. M. A. M. van: Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise.J. Math. Kyoto Univ. 43 (2003), 261-303. MR 2051026, 10.1215/kjm/1250283728 |
| Reference:
|
[6] Brze'zniak, Z., Zabczyk, J.: Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise.Potential Anal. 32 (2010), 153-188. MR 2584982, 10.1007/s11118-009-9149-1 |
| Reference:
|
[7] Caithamer, P.: The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise.Stoch. Dyn. 5 (2005), 45-64. Zbl 1083.60053, MR 2118754, 10.1142/S0219493705001286 |
| Reference:
|
[8] Carmona, R., Fouque, J.-P., Vestal, D.: Interacting particle systems for the computation of rare credit portfolio losses.Finance Stoch. 13 (2009), 613-633. MR 2519846, 10.1007/s00780-009-0098-8 |
| Reference:
|
[9] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, Vol. 44.Cambridge University Press Cambridge (2008). MR 1207136 |
| Reference:
|
[10] Decreusefond, L., Üstünel, A. S.: Stochastic analysis of the fractional Brownian motion.Potential Anal. 10 (1999), 177-214. Zbl 0924.60034, MR 1677455, 10.1023/A:1008634027843 |
| Reference:
|
[11] Dettweiler, J., Weis, L., Neerven, J. M. A. M. van: Space-time regularity of solutions of the parabolic stochastic Cauchy problem.Stochastic Anal. Appl. 24 (2006), 843-869. MR 2241096, 10.1080/07362990600753577 |
| Reference:
|
[12] Duncan, T. E., Pasik-Duncan, B., Maslowski, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces.Stoch. Dyn. 2 (2002), 225-250. Zbl 1040.60054, MR 1912142, 10.1142/S0219493702000340 |
| Reference:
|
[13] Fernique, X.: Intégrabilité des vecteurs gaussiens.C. R. Acad. Sci. Paris, Sér. A 270 (1970), 1698-1699 French. Zbl 0206.19002, MR 0266263 |
| Reference:
|
[14] Feyel, D., Pradelle, A. de La: On fractional Brownian processes.Potential Anal. 10 (1999), 273-288. Zbl 0944.60045, MR 1696137, 10.1023/A:1008630211913 |
| Reference:
|
[15] Goldys, B., Neerven, J. M. A. M. van: Transition semigroups of Banach space valued Ornstein-Uhlenbeck processes.Acta Appl. Math. 76 (2003), 283-330 Revised version: arXiv:math/0606785. MR 1976297, 10.1023/A:1023261101091 |
| Reference:
|
[16] Grecksch, W., Roth, C., Anh, V. V.: $Q$-fractional Brownian motion in infinite dimensions with application to fractional Black-Scholes market.Stochastic Anal. Appl. 27 (2009), 149-175. Zbl 1158.60364, MR 2473144, 10.1080/07362990802565084 |
| Reference:
|
[17] Guasoni, P.: No arbitrage under transaction costs, with fractional Brownian motion and beyond.Math. Finance 16 (2006), 569-582. Zbl 1133.91421, MR 2239592, 10.1111/j.1467-9965.2006.00283.x |
| Reference:
|
[18] Gubinelli, M., Lejay, A., Tindel, S.: Young integrals and SPDEs.Potential Anal. 25 (2006), 307-326. Zbl 1103.60062, MR 2255351, 10.1007/s11118-006-9013-5 |
| Reference:
|
[19] Hairer, M., Ohashi, A.: Ergodic theory for SDEs with extrinsic memory.Ann. Probab. 35 (2007), 1950-1977. Zbl 1129.60052, MR 2349580, 10.1214/009117906000001141 |
| Reference:
|
[20] Hu, Y.: Heat equations with fractional white noise potentials.Appl. Math. Optimization 43 (2001), 221-243. Zbl 0993.60065, MR 1885698, 10.1007/s00245-001-0001-2 |
| Reference:
|
[21] Hu, Y., Øksendal, B., Zhang, T.: General fractional multiparameter white noise theory and stochastic partial differential equations.Commun. Partial. Diff. Equations 29 (2004), 1-23. MR 2038141, 10.1081/PDE-120028841 |
| Reference:
|
[22] Jumarie, G.: Would dynamic systems involving human factors be necessarily of fractal nature?.Kybernetes 31 (2002), 1050-1058. Zbl 1113.37316, 10.1108/03684920210436336 |
| Reference:
|
[23] Jumarie, G.: New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations.Math. Comput. Modelling 44 (2006), 231-254. Zbl 1130.92043, MR 2239054, 10.1016/j.mcm.2005.10.003 |
| Reference:
|
[24] Kalton, N. J., Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Embedding vector-valued Besov spaces into spaces of gamma-radonifying operators.Math. Nachr. 281 (2008), 238-252. MR 2387363, 10.1002/mana.200510598 |
| Reference:
|
[25] Kalton, N. J., Weis, L.: The {$H^\infty$}-calculus and square function estimates.In preparation. Zbl 1111.47020 |
| Reference:
|
[26] Leland, W., Taqqu, M., Willinger, W., Wilson, D.: On the self-similar nature of ethernet traffic.IEEE/ACM Trans. Networking 2 (1994), 1-15. 10.1109/90.282603 |
| Reference:
|
[27] Mandelbrot, B. B., Ness, J. W. Van: Fractional Brownian motion, fractional noises, and applications.SIAM Rev. 10 (1968), 422-437. MR 0242239, 10.1137/1010093 |
| Reference:
|
[28] Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion.J. Funct. Anal. 202 (2003), 277-305. Zbl 1027.60060, MR 1994773, 10.1016/S0022-1236(02)00065-4 |
| Reference:
|
[29] Maslowski, B., Schmalfuss, B.: Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion.Stochastic Anal. Appl. 22 (2004), 1577-1607. Zbl 1062.60060, MR 2095071, 10.1081/SAP-200029498 |
| Reference:
|
[30] Neerven, J. M. A. M. van: $\gamma$-radonifying operators---a survey.Proceedings CMA 44 (2010), 1-61. MR 2655391 |
| Reference:
|
[31] Neerven, J. M. A. M. van: Stochastic Evolution Equations. Lecture Notes of the Internet Seminar 2007/08, OpenCourseWare, TU Delft, http://ocw.tudelft.nl.. |
| Reference:
|
[32] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic integration in UMD Banach spaces.Ann. Probab. 35 (2007), 1438-1478. MR 2330977, 10.1214/009117906000001006 |
| Reference:
|
[33] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Conditions for stochastic integrability in UMD Banach spaces.Inn: Banach Spaces and their Applications in Analysis: In Honor of Nigel Kalton's 60th Birthday. De Gruyter Proceedings in Mathematics Walter De Gruyter Berlin (2007), 127-146. MR 2374704 |
| Reference:
|
[34] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic evolution equations in UMD Banach spaces.J. Funct. Anal. 255 (2008), 940-993. MR 2433958, 10.1016/j.jfa.2008.03.015 |
| Reference:
|
[35] Neerven, J. M. A. M. van, Weis, L.: Stochastic integration of functions with values in a Banach space.Stud. Math. 166 (2005), 131-170. MR 2109586, 10.4064/sm166-2-2 |
| Reference:
|
[36] Nualart, D.: Fractional Brownian motion: stochastic calculus and applications.In: Proceedings of the International Congress of Mathematicians. Volume III: Invited Lectures, Madrid, Spain, August 22-30, 2006 European Mathematical Society Zürich (2006), 1541-1562. Zbl 1102.60033, MR 2275741 |
| Reference:
|
[37] Ohashi, A.: Fractional term structure models: no-arbitrage and consistency.Ann. Appl. Probab. 19 (2009), 1553-1580. Zbl 1188.91229, MR 2538080, 10.1214/08-AAP586 |
| Reference:
|
[38] Palmer, T. N., Shutts, G. J., Hagedorn, R., Doblas-Reyes, F. J., Jung, T., Leutbecher, M.: Representing model uncertainty in weather and climate prediction.Annu. Rev. Earth Planet. Sci. 33 (2005), 163-193. MR 2153320, 10.1146/annurev.earth.33.092203.122552 |
| Reference:
|
[39] Palmer, T. N.: A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models.Q. J. Meteorological Soc. 127 (2001) B 279-304. |
| Reference:
|
[40] Pasik-Duncan, B., Duncan, T. E., Maslowski, B.: Linear stochastic equations in a Hilbert space with a fractional Brownian motion.In: Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems. International Series in Operations Research & Management Science Springer New York (2006), 201-221. Zbl 1133.60015, MR 2353483 |
| Reference:
|
[41] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44.Springer New York (1983). MR 0710486 |
| Reference:
|
[42] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications.Gordon and Breach New York (1993). Zbl 0818.26003, MR 1347689 |
| Reference:
|
[43] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Math. Library, vol. 18.North-Holland Amsterdam (1978). MR 0503903 |
| Reference:
|
[44] Tindel, S., Tudor, C. A., Viens, F.: Stochastic evolution equations with fractional Brownian motion.Probab. Theory Relat. Fields 127 (2003), 186-204. Zbl 1036.60056, MR 2013981, 10.1007/s00440-003-0282-2 |
| Reference:
|
[45] Weis, L.: Operator-valued Fourier multiplier theorems and maximal $L\sb p$-regularity.Math. Ann. 319 (2001), 735-758. MR 1825406, 10.1007/PL00004457 |
| Reference:
|
[46] Willinger, W., Taqqu, M., Leland, W. E., Wilson, D. V.: Self-similarity in high speed packet traffic: analysis and modeling of Ethernet traffic measurements.Stat. Sci. 10 (1995), 67-85. Zbl 1148.90310, 10.1214/ss/1177010131 |
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