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The Feynman–Kac Formula and Excursion Theory

  • Ju-Yi Yen
  • Marc Yor
Chapter
  • 1.5k Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2088)

Abstract

We provide a proof of the Feynman–Kac formula for Brownian motion, using excursion theory up to an independent exponential time θ. Call g(θ) the last zero before θ. The independence of the pre-g(θ) process and the post-g(θ) process and the representation of their laws in terms of the integrals of Wiener measure up to inverse local time, or first hitting times allow to recover a formulation of the Feynman–Kac formula via excursion theory.

Keywords

Excursion Theory Independent Exponential Time Inverse Local Time Wiener Measure Brownian Motion 
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References

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    M. Kac, On the average of a certain Wiener functional and a related limit theorem in calculus of probability. Trans. Am. Math. Soc. 65, 401–414 (1946)CrossRefGoogle Scholar
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    M. Kac, On the distribution of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)CrossRefzbMATHGoogle Scholar
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    M. Kac, On some connections between probability theory and differential and integral equations, in Proc. Second Berkeley Symp. Math. Stat. Prob., ed. by J. Neyman (University of California Press, Berkeley, 1951), pp. 189–215Google Scholar
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    M. Jeanblanc, J. Pitman, M. Yor, The Feynman–Kac formula and decomposition of Brownian paths. Sociedade Brasiliera de Matemática Applicada e Computacional. Matemática Aplicada e Computacional. Comput. Appl. Math. 16(1), 27–52 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Ju-Yi Yen
    • 1
  • Marc Yor
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris VIParis CX 05France

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