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An Approach to Stochastic Integration in General Separable Banach Spaces

  • A. A. Kalinichenko
Article
  • 44 Downloads

Abstract

We suggest a new approach to stochastic integration in infinite-dimensional spaces that is based on representing random variables on Banach spaces as real-valued processes on an interval. We prove stochastic integrability of operator-valued processes on general separable Banach spaces under the conditions that do not depend on the norm of the space and show how our methods can be applied to studying infinite-dimensional stochastic differential equations. In particular, our results provide a natural construction of the stochastic integral in abstract Wiener spaces.

Keywords

Infinite-dimensional stochastic analysis Stochastic integral Stochastic differential equations Gaussian measures 

Mathematics Subject Classification (2010)

60H05 60G15 60H10 

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References

  1. 1.
    Bogachev, V., Smolyanov, O.: Topological Vector Spaces and their Applications, Springer (2017)Google Scholar
  2. 2.
    Bogachev, V.I.: Gaussian Measures, vol. 62. AMS (1998).  https://doi.org/10.1090/surv/062
  3. 3.
    Brooks, J., Dinculeanu, N.: Stochastic integration in banach spaces. Adv. Math. 81(1), 99–104 (1990).  https://doi.org/10.1016/0001-8708(90)90006-9 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brzeźniak, Z., Carroll, A.: Approximations of the wong–zakai type for stochastic differential equations in m-type 2 banach spaces with applications to loop spaces. In: Séminaire de Probabilités XXXVII, pp 251–289. Springer, Berlin (2003),  https://doi.org/10.1007/978-3-540-40004-2_11
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014).  https://doi.org/10.1017/CBO9781107295513 CrossRefzbMATHGoogle Scholar
  6. 6.
    Dettweiler, E.: Banach space valued processes with independent increments and stochastic integration. In: Probability in Banach Spaces IV, pp 54–83 (1983),  https://doi.org/10.1007/bfb0064263
  7. 7.
    Di Girolami, C., Fabbri, G., Russo, F.: The covariation for banach space valued processes and applications. Metrika 77(1), 51–104 (2014).  https://doi.org/10.1007/s00184-013-0472-6 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dinculeanu, N.: Vector Integration and Stochastic Integration in Banach Spaces, vol. 48. Wiley, New York (2000).  https://doi.org/10.1002/9781118033012 CrossRefzbMATHGoogle Scholar
  9. 9.
    Kallenberg, O.: Foundations of modern probability. Springer, Berlin (2006).  https://doi.org/10.1007/b98838 zbMATHGoogle Scholar
  10. 10.
    Kelley, J.L.: General Topology. Springer, Berlin (1975)zbMATHGoogle Scholar
  11. 11.
    Kuo, H.H.: Stochastic integrals in abstract wiener space. Pac. J. Math. 41(2), 469–483 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991).  https://doi.org/10.1007/978-3-642-20212-4 CrossRefzbMATHGoogle Scholar
  13. 13.
    Metivier, M., Pellaumail, J.: Stochastic Integration. Probability and Mathematical Statistics. Academic, New York (1980)zbMATHGoogle Scholar
  14. 14.
    Neerven, J.V., Veraar, M., Weis, L.: Stochastic integration in banach spaces–a survey. In: Stochastic Analysis: A Series of Lectures, pp 297–332. Springer, Berlin (2015).  https://doi.org/10.1007/978-3-0348-0909-2_11
  15. 15.
    Neerven, J.V., Weis, L.: Stochastic integration of operator-valued functions with respect to Banach space-valued Brownian motion. Potential Anal. 29(1), 65–88 (2008).  https://doi.org/10.1007/s11118-008-9088-2 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ondreját, M.: Integral representations of cylindrical local martingales in every separable banach space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(03), 365–379 (2007).  https://doi.org/10.1142/S0219025707002816 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Talagrand, M.: Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems, vol. 60. Springer, Berlin (2014).  https://doi.org/10.1007/978-3-642-54075-2 CrossRefzbMATHGoogle Scholar
  18. 18.
    Teichmann, J.: Another approach to some rough and stochastic partial differential equations. Stochastics Dyn. 11(02n03), 535–550 (2011).  https://doi.org/10.1142/S0219493711003437 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vakhania, N., Tarieladze, V., Chobanyan, S.: Probability Distributions on Banach Spaces, vol. 14. Springer, Berlin (1987).  https://doi.org/10.1007/978-94-009-3873-1 CrossRefGoogle Scholar
  20. 20.
    Veraar, M., Yaroslavtsev, I.: Cylindrical continuous martingales and stochastic integration in infinite dimensions. Electron. J. Probab. 21 (2016).  https://doi.org/10.1214/16-EJP7

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyMoscowRussia

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