Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution
Pure and Applied Mathematics Journal
Volume 4, Issue 3, June 2015, Pages: 133-138
Received: Dec. 15, 2014; Accepted: Dec. 16, 2014; Published: Jun. 11, 2015
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Author
Badri Mamporia, Niko Muskhelishvili Institute of Computational Mathematics, Technical University of Georgia, Tbilisi, Georgia
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Abstract
In this paper the stochastic differential equation in a Banach space is considered for the case when the Wiener process in the equation is Banach space valued and the integrand non-anticipating function is operator-valued. At first the stochastic differential equation for the generalized random process is introduced and developed existence and uniqueness of solutions as the generalized random process. The corresponding results for the stochastic differential equation in a Banach space is given. In [5] we consider the stochastic differential equation in a Banach space in the case, when the Wiener process is one dimensional and the integrand non-anticipating function is Banach space valued.
Keywords
Covariance Operators, Ito Stochastic Integrals and Stochastic Differential Equations in a Banach Space, Wiener Process in a Banach Space
To cite this article
Badri Mamporia, Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution, Pure and Applied Mathematics Journal. Vol. 4, No. 3, 2015, pp. 133-138. doi: 10.11648/j.pamj.20150403.22
References
[1]
McConnell T.R. Decoupling and stochastic integration in UMD Banach spaces. Probab. Math. Statist., 1989, v. 10, No. 2, p. 283-295.
[2]
Da Prato G., Zabczyk J. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press,1992.
[3]
Brzezniak Z., van Neerven J. M.A.M., Veraar M.C., Weis L. Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differential Equations, 2008, v. 245, p. 30-58.
[4]
Rosinski J,. Suchanecki Z. On the space of vector-valued functions integrable with respect to the white noise. Colloq. Math., 1980, v. 43, No. 1, p. 183-201.
[5]
B. Mamporia. Stochastic differential equation for generalized random processes in a Banach space. Theory of probability and its Applications, 56(4),602-620,2012, SIAM.Teoriya Veroyatnostei i ee Primeneniya, 56:4 (2011), 704-725.
[6]
Vakhania N.N., Ta¬rieladze V.I., Chobanyan S.A. Probability dis¬tri¬bu¬tions on Ba¬nach spa¬¬ces. D. Reidel, 1987.
[7]
N. N. Vakhania, Probability distributions on linear spa¬ces. North Hol¬land, 1981.
[8]
B.Mamporia. Wiener Processes and Stochastic Integrals in a Banach space. Probability and Mathematical Statistics, Vol. 7, Fasc. 1 (1986), p.59-75.
[9]
B. Mamporia . On Wiener process in a Frechet space. Soobshch. Acad. Nauk Gruzin. SSR, 1977.
[10]
S. Chevet. Seminaire sur la geometrie des espaces de Banach, Ecole Politechnique, Centre de Mathematique, Exp. . XIX, 1977- 1978.
[11]
Nguen Van Thu, Banach space valued Brownian motions, Acta Math. Vietnamica 3 (2) (1978), p. 35-46.
[12]
Kwapien and B. Szymanski. Some remarks on Gaussian measures on Banach space. Probab. Math. Statist. 1(1980), No.1, p. 59-65.
[13]
B. Mamporia. On the Ito formula in a Banach space. Georgian Mathematical Journal Vol.7 No1, p. 155-168.
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