Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution

Badri Mamporia

doi:doi:10.11648/j.pamj.20150403.22

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Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution

Badri Mamporia

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 3)

Received: 15 December 2014 Accepted: 16 December 2014 Published: 11 June 2015

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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.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 3)
DOI	10.11648/j.pamj.20150403.22
Page(s)	133-138
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Covariance Operators, Ito Stochastic Integrals and Stochastic Differential Equations in a Banach Space, Wiener Process in a Banach Space

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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APA Style

Badri Mamporia. (2015). Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution. Pure and Applied Mathematics Journal, 4(3), 133-138. https://doi.org/10.11648/j.pamj.20150403.22

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ACS Style

Badri Mamporia. Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution. Pure Appl. Math. J. 2015, 4(3), 133-138. doi: 10.11648/j.pamj.20150403.22

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AMA Style

Badri Mamporia. Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution. Pure Appl Math J. 2015;4(3):133-138. doi: 10.11648/j.pamj.20150403.22

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@article{10.11648/j.pamj.20150403.22,
  author = {Badri Mamporia},
  title = {Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {3},
  pages = {133-138},
  doi = {10.11648/j.pamj.20150403.22},
  url = {https://doi.org/10.11648/j.pamj.20150403.22},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150403.22},
  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.},
 year = {2015}
}

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TY  - JOUR
T1  - Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution
AU  - Badri Mamporia
Y1  - 2015/06/11
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150403.22
DO  - 10.11648/j.pamj.20150403.22
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JO  - Pure and Applied Mathematics Journal
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EP  - 138
PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.pamj.20150403.22
AB  - 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.
VL  - 4
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Author Information

Badri Mamporia

Niko Muskhelishvili Institute of Computational Mathematics, Technical University of Georgia, Tbilisi, Georgia

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APA Style

Badri Mamporia. (2015). Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution. Pure and Applied Mathematics Journal, 4(3), 133-138. https://doi.org/10.11648/j.pamj.20150403.22

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ACS Style

Badri Mamporia. Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution. Pure Appl. Math. J. 2015, 4(3), 133-138. doi: 10.11648/j.pamj.20150403.22

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AMA Style

Badri Mamporia. Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution. Pure Appl Math J. 2015;4(3):133-138. doi: 10.11648/j.pamj.20150403.22

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@article{10.11648/j.pamj.20150403.22,
  author = {Badri Mamporia},
  title = {Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {3},
  pages = {133-138},
  doi = {10.11648/j.pamj.20150403.22},
  url = {https://doi.org/10.11648/j.pamj.20150403.22},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150403.22},
  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.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Stochastic differential equation driven by the Wiener process in a Banach space, existence and uniqueness of the generalized solution
AU  - Badri Mamporia
Y1  - 2015/06/11
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150403.22
DO  - 10.11648/j.pamj.20150403.22
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 133
EP  - 138
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150403.22
AB  - 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.
VL  - 4
IS  - 3
ER  -

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