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Decomposition of Brownian motion with derivation in a local minimum by the juxtaposition of its positive and negative excursions. (Décomposition du mouvement brownien avec dérive en un minimum local par juxtaposition de ses excursions positives et négatives.) (French) Zbl 0741.60077
Séminaire de probabilités, Lect. Notes Math. 1485, 330-344 (1991).
[For the entire collection see Zbl 0733.00018.]
The central result of the paper is an identity in law between the following two pairs of processes. The first one is obtained after decomposing a Brownian motion with constant drift \(X=(X_ s:\;s\in[0,1])\) at the instant of its absolute minimum. The second pair is obtained by the ‘juxtaposition’ respectively of the positive and of the negative excursions of X. Specifically, the process constructed from the positive excursions is \((X+{1\over 2}L)\circ\alpha^ +_ .\), where \(L\) is the Tanaka local time at 0 and \(\alpha^ +_ .\) the inverse of the time spent by \(X\) in the positive half-line. As a particular case of the identity, the preceding process has the same law as the post-minimum process. The proof relies on elementary excursion theory. I. Karatzas and S. E. Shreve [Stat. Probab. Lett. 5, 87-93 (1987; Zbl 0615.60075)] have a related result for Brownian motion with zero drift.
W. Vervaat [Ann. Probab. 7, 143-149 (1979; Zbl 0392.60058)] gave the following construction of the normalized Brownian excursion [see also Ph. Biane, Ann. Inst. Henri Poincaré, Probab. Stat. 22, 1-7 (1986; Zbl 0596.60079)]. Take a Brownian bridge, split it at its absolute minimum, and past the pre-minimum part at the end of the post-minimum part. The resulting process is a normalized Brownian excursion. Here, an application of the main identity yields the following relation. Take a Brownian bridge, time-reverse the pre-minimum part, and then past the post-minimum part. The resulting process is a Brownian meander. The same path-transformation applied to a Brownian motion gives a three- dimensional Bessel process.
Reviewer: J.Bertoin (Paris)

60J65 Brownian motion
60J55 Local time and additive functionals
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