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Stochastic Differential Equations and Stochastic Flows

  • Hiroshi Kunita
Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 92)

Abstract

In this chapter, we show that solutions of a continuous symmetric stochastic differential equation (SDE) on a Euclidean space define a continuous stochastic flow of diffeomorphisms and that solutions of an SDE with diffeomorphic jumps define a right continuous stochastic flow of diffeomorphisms. Sections 3.1 and 3.2 are introductions. Definitions of these SDEs and stochastic flows will be given and the geometric property of solutions will be explained as well as basic facts. Rigorous proof of these facts will be given in Sects. 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9. In Sect. 3.3, we study another Itô SDE with parameter, called the master equation. Applying results of Sect. 3.3, we show in Sect. 3.4 that solutions of the original SDE define a stochastic flow of C-maps. For the proof of the diffeomorphic property, we need further arguments. In Sect. 3.5 we consider backward SDE and backward stochastic flow of C-maps. Further, the forward–backward calculus for stochastic flow will be discussed in Sects. 3.5, 3.6 and 3.8. These facts will be applied in Sects. 3.7 and 3.9 for proving the diffeomorphic property of solutions.

References

  1. 8.
    Bismut, J.M.: Mécanique Aléatoire. Lecture Notes in Mathematics, vol. 866. Springer, Berlin/Heidelberg/New York (1981)Google Scholar
  2. 14.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)Google Scholar
  3. 25.
    Elworthy, K.D.: Stochastic Differential Equations on Manifolds. LMS Lecture Note Series, vol. 70. Cambridge University Press, Cambridge (1982)Google Scholar
  4. 30.
    Fujiwara, T., Kunita, H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25, 71–106 (1985)MathSciNetCrossRefGoogle Scholar
  5. 31.
    Fujiwara, T., Kunita, H.: Canonical SDE’s based on semimartingales with spatial parameters. Part II; Inverse flows and backward SDE’s. Kyushu J. Math. 53, 301–331 (1999)Google Scholar
  6. 36.
    Harris, T.E.: Coalescing and noncoalescing stochastic flows in R 1. Stoch. Proc. Appl. 17, 187–210 (1984)MathSciNetCrossRefGoogle Scholar
  7. 41.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North Holland, Amsterdam (1989)zbMATHGoogle Scholar
  8. 50.
    Itô, K.: On stochastic differential equations. Memoirs Am. Math. Soc. 4, 1–51 (1951)Google Scholar
  9. 59.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  10. 60.
    Kunita, H.: Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In: Rao, M.M. (ed.) Real and Stochastic Analysis. Birkhäuser, Boston (2004)zbMATHGoogle Scholar
  11. 75.
    Le Jan, Y.: Flots de diffusions dans \({\mathbb R}^d\). C.R. Acad. Sci. Paris, Ser. I 294, 697–699 (1982)Google Scholar
  12. 78.
    Malliavin, P.: Géometrie Différentielle Stochastique. Les Presses de l’Université, Montréal (1978)zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Hiroshi Kunita
    • 1
  1. 1.Kyushu University (emeritus)FukuokaJapan

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