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Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. (English) Zbl 0877.60053

Ikeda, N. (ed.) et al., Itô’s stochastic calculus and probability theory. Tribute dedicated to Kiyosi Itô on the occasion of his 80th birthday. Tokyo: Springer. 293-310 (1996).
The double description of Itô’s excursion measure, by conditioning either on the lifetime or on the maximum, is classical in the real Brownian case. It gives an identity, called “agreement formula”, between two measures on excursion paths, that involves three-dimensional Bessel processes (and the corresponding bridges). The generalization of this formula involving Bessel processes of any dimension is also known. The main aim of the present article is to generalize again this agreement formula, letting it involves a very general real diffusion \(X_t\). The proof is based on the (already known) William’s decomposition at the maximum of \(X_t\), with an explicit expression for the joint law of \((X_t\), Max \(X[0,t]\), Argmax \(X[0,t])\). Several consequences are derived, mainly relative to the hitting time of 1 by the Bessel process of dimension \(\delta\). The whole article is also intended to be a survey around the agreement formula.
For the entire collection see [Zbl 0852.00016].
Reviewer: J.Franchi (Paris)

MSC:

60J60 Diffusion processes
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