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An Itō Formula in the Space of Tempered Distributions

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Abstract

We extend the Itō formula (Rajeev in From Tanaka’s formula to Ito’s formula: distributions, tensor products and local times, Springer, Berlin, 2001, Theorem 2.3) for semimartingales with paths that are right continuous and have left limits. We also comment on the local time process of such semimartingales. We apply the Itō formula to Lévy processes to obtain existence of solutions to certain classes of stochastic differential equations in the Hermite–Sobolev spaces.

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  • 01 November 2017

    The following corrections are required in Theorem 4.7.

References

  1. Applebaum, D.: Lévy Processes and Stochastic Calculus: Cambridge Studies in Advanced Mathematics, vol. 116, 2nd edn. Cambridge University Press, Cambridge (2009). doi:10.1017/CBO9780511809781

    Book  Google Scholar 

  2. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions: Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992). doi:10.1017/CBO9780511666223

    Book  MATH  Google Scholar 

  3. Hida, T.: Brownian motion. Applications of Mathematics, vol. 11 (Translated from the Japanese by the author and T. P. Speed). Springer, New York (1980)

  4. Itō, K.: Foundations of stochastic differential equations in infinite-dimensional spaces. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 47. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1984)

  5. Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, second edn. Springer, Berlin (2003)

  6. Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York). Springer, New York (1997)

    MATH  Google Scholar 

  7. Kallianpur, G., Xiong, J.: Stochastic differential equations in infinite-dimensional spaces. Institute of Mathematical StatisticsLecture Notes—Monograph Series, 26. Institute of MathematicalStatistics, Hayward, CA (1995). Expanded version of the lectures delivered as part of the 1993 Barrett Lectures at the University of Tennessee, Knoxville, TN, March 25–27, 1993, With a foreword by Balram S. Rajput and Jan Rosinski

  8. Kunita, H.: Stochastic integrals based on martingales taking values in Hilbert space. Nagoya Math. J. 38, 41–52 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  9. Métivier, M.: Semimartingales. de Gruyter Studies in Mathematics, vol. 2. Walter de Gruyter & Co., Berlin (1982). A course on stochastic processes

  10. Métivier, M., Pellaumail, J.: Stochastic Integration. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1980). Probability and Mathematical Statistics

    MATH  Google Scholar 

  11. Mitoma, I.: Martingales of random distributions. Mem. Fac. Sci. Kyushu Univ. Ser. A 35(1), 185–197 (1981). doi:10.2206/kyushumfs.35.185

    MATH  MathSciNet  Google Scholar 

  12. Protter, P.E.: Stochastic integration and differential equations. Applications of Mathematics (New York), vol. 21, second edn. Springer, Berlin (2004). Stochastic Modelling and Applied Probability

  13. Rajeev, B.: From Tanaka’s formula to Ito’s formula: distributions, tensor products and local times. In: Séminaire de Probabilités, XXXV, Lecture Notes in Math., vol. 1755, pp. 371–389. Springer, Berlin (2001)

  14. Rajeev, B.: From Tanaka’s formula to Itô’s formula: the fundamental theorem of stochastic calculus. Proc. Indian Acad. Sci. Math. Sci. 107(3), 319–327 (1997). doi:10.1007/BF02867261

    Article  MATH  MathSciNet  Google Scholar 

  15. Rajeev, B.: Translation invariant diffusion in the space of tempered distributions. Indian J. Pure Appl. Math. 44(2), 231–258 (2013). doi:10.1007/s13226-013-0012-0

    Article  MATH  MathSciNet  Google Scholar 

  16. Rajeev, B., Thangavelu, S.: Probabilistic representations of solutions to the heat equation. Proc. Indian Acad. Sci. Math. Sci. 113(3), 321–332 (2003). doi:10.1007/BF02829609

    Article  MATH  MathSciNet  Google Scholar 

  17. Rajeev, B., Thangavelu, S.: Probabilistic representations of solutions of the forward equations. Potential Anal. 28(2), 139–162 (2008). doi:10.1007/s11118-007-9074-0

    Article  MATH  MathSciNet  Google Scholar 

  18. Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Dover Publications Inc., Mineola (2006). Unabridged republication of the 1967 original

    MATH  Google Scholar 

  19. Üstünel, A.S.: A generalization of Itô’s formula. J. Funct. Anal. 47(2), 143–152 (1982). doi:10.1016/0022-1236(82)90102-1

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author would like to thank Professor B. Rajeev, Indian Statistical Institute, Bangalore, for valuable suggestions during the work and pointing out the way to Theorem 4.5.

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  1. Indian Statistical Institute Bangalore Centre, Bangalore, India

    Suprio Bhar

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  1. Suprio Bhar
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Correspondence to Suprio Bhar.

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A correction to this article is available online at https://doi.org/10.1007/s10959-017-0792-y.

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Bhar, S. An Itō Formula in the Space of Tempered Distributions. J Theor Probab 30, 510–528 (2017). https://doi.org/10.1007/s10959-015-0639-3

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  • DOI: https://doi.org/10.1007/s10959-015-0639-3

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