The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space

Badri Mamporia

doi:doi:10.11648/j.pamj.20150404.15

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The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space

Badri Mamporia

Received: 30 May 2015 Accepted: 12 June 2015 Published: 10 August 2015

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Abstract

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

DOI	10.11648/j.pamj.20150404.15
Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 4, August 2015)
Page(s)	164-171
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), 2024. Published by Science Publishing Group

Keywords

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

References

[1]	I. I. Belopolskaia, Yu. L. Daletzky (1978), Diffussion Process in Smooth Banach Spaces and Manifolds. (Russian) Trudy Moskov. Mat. Obshch. 37, p.107–141.
[2]	Z.Brzez’niak, J.M.A.M. van Neerven, M.C. Veraar, L Weis (2008), Ito’s Formula in UMD Banach Spaces and Regularity of Solutions of the Zakai Equation. Journal of Differential Equations v.245, 1, p.30-58.
[3]	B. Mamporia (2000), On the Ito Formula in a Banach Space. Georgian Mathematical Journal vol.7, 1, p.155-168.
[4]	B. Mamporia, Ito’s formula in a Banach space. Bull. Georgian National Academy of Sciences, vol. 5, no. 3, 2011, 12-16.
[5]	N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanian (1985), Probability Distributions on Banach Spaces. Nauka, Moskow; The English translation: Reidel, Dordrecht, the Netherlands, 1987. p.482.
[6]	B. Mamporia, On Wiener process in a Frechet space. Soobshch. Acad. Nauk Gruzin. SSR, 1977
[7]	B. Mamporia, Wiener Processes and Stochastic Integrals in a Banach space. Probability and Ma thematical Statistics, Vol. 7, Fasc. 1 (1986), p.59-75.
[8]	S. Kwapie´n and B. Szymanski, Some remarks on Gaussian measures on Banach space. Probab. Math. Statist. 1(1980), No. 1, p. 59–65
[9]	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.
[10]	H.H. Kuo, Gaussian measures in Banach spaces. Springer- Verlag, Berlin, Heidelberg, New York, 1975.
[11]	Da Prato G., Zabczyk J. Stochastic Differential Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Aplications. Cambridge University Press, 1992.
[12]	B. 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 2015:4(3):133-138.

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Badri Mamporia. (2015). The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure and Applied Mathematics Journal, 4(4), 164-171. https://doi.org/10.11648/j.pamj.20150404.15

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ACS Style

Badri Mamporia. The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure Appl. Math. J. 2015, 4(4), 164-171. doi: 10.11648/j.pamj.20150404.15

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AMA Style

Badri Mamporia. The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure Appl Math J. 2015;4(4):164-171. doi: 10.11648/j.pamj.20150404.15

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@article{10.11648/j.pamj.20150404.15,
  author = {Badri Mamporia},
  title = {The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {4},
  pages = {164-171},
  doi = {10.11648/j.pamj.20150404.15},
  url = {https://doi.org/10.11648/j.pamj.20150404.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150404.15},
  abstract = {Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.},
 year = {2015}
}

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TY  - JOUR
T1  - The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space
AU  - Badri Mamporia
Y1  - 2015/08/10
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150404.15
DO  - 10.11648/j.pamj.20150404.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 164
EP  - 171
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150404.15
AB  - Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.
VL  - 4
IS  - 4
ER  -

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Author Information

Badri Mamporia

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

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Plain Text BibTeX RIS

APA Style

Badri Mamporia. (2015). The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure and Applied Mathematics Journal, 4(4), 164-171. https://doi.org/10.11648/j.pamj.20150404.15

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ACS Style

Badri Mamporia. The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure Appl. Math. J. 2015, 4(4), 164-171. doi: 10.11648/j.pamj.20150404.15

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AMA Style

Badri Mamporia. The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure Appl Math J. 2015;4(4):164-171. doi: 10.11648/j.pamj.20150404.15

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@article{10.11648/j.pamj.20150404.15,
  author = {Badri Mamporia},
  title = {The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {4},
  pages = {164-171},
  doi = {10.11648/j.pamj.20150404.15},
  url = {https://doi.org/10.11648/j.pamj.20150404.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150404.15},
  abstract = {Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space
AU  - Badri Mamporia
Y1  - 2015/08/10
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150404.15
DO  - 10.11648/j.pamj.20150404.15
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 164
EP  - 171
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150404.15
AB  - Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.
VL  - 4
IS  - 4
ER  -

Copy | Download