Abstract
Following the previous Chap. 5, two disintegrations of the Itô measure n with respect to the lifetime and the maximum are stated and later proven. The first of these, due to Itô, features the BES(3) bridges, whereas the second one, due to Williams, features two BES(3) processes put back to back, and considered up to their first hitting time of a level. That these two descriptions hold jointly translates into an agreement formula between BES(3) bridges and two BES(3) processes put back to back. Finally, n is shown to be Markovian, its entrance law is described, and it is also shown that the corresponding semigroup is that of Brownian motion killed as it reaches 0.
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References
L.C.G. Rogers, Williams’ characterisation of the Brownian excursion law: proof and applications. Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French). Lecture Notes in Math., vol. 850. (Springer, Berlin, 1981), pp. 227–250
D. Revuz, M. Yor, Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. (Springer, Berlin, 1999)
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Yen, JY., Yor, M. (2013). Two Descriptions of n: Itô’s and Williams’. In: Local Times and Excursion Theory for Brownian Motion. Lecture Notes in Mathematics, vol 2088. Springer, Cham. https://doi.org/10.1007/978-3-319-01270-4_6
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DOI: https://doi.org/10.1007/978-3-319-01270-4_6
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