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An Approach to Stochastic Integration in General Separable Banach Spaces

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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.

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References

  1. Bogachev, V., Smolyanov, O.: Topological Vector Spaces and their Applications, Springer (2017)

  2. Bogachev, V.I.: Gaussian Measures, vol. 62. AMS (1998). https://doi.org/10.1090/surv/062

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014). https://doi.org/10.1017/CBO9781107295513

    Book  MATH  Google Scholar 

  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. 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

    Article  MathSciNet  MATH  Google Scholar 

  8. Dinculeanu, N.: Vector Integration and Stochastic Integration in Banach Spaces, vol. 48. Wiley, New York (2000). https://doi.org/10.1002/9781118033012

    Book  MATH  Google Scholar 

  9. Kallenberg, O.: Foundations of modern probability. Springer, Berlin (2006). https://doi.org/10.1007/b98838

    MATH  Google Scholar 

  10. Kelley, J.L.: General Topology. Springer, Berlin (1975)

    MATH  Google Scholar 

  11. Kuo, H.H.: Stochastic integrals in abstract wiener space. Pac. J. Math. 41(2), 469–483 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Book  MATH  Google Scholar 

  13. Metivier, M., Pellaumail, J.: Stochastic Integration. Probability and Mathematical Statistics. Academic, New York (1980)

    MATH  Google Scholar 

  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. 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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Book  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Book  Google Scholar 

  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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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Moscow, Russia

    A. A. Kalinichenko

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  1. A. A. Kalinichenko
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Kalinichenko, A.A. An Approach to Stochastic Integration in General Separable Banach Spaces. Potential Anal 50, 591–608 (2019). https://doi.org/10.1007/s11118-018-9696-4

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