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Amplitude of the Brownian motion and juxtaposition of positive and negative excursions. (Amplitude du mouvement brownien et juxtaposition des excursions positives et négatives.) (French) Zbl 0763.60038
Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 361-373 (1992).
[For the entire collection see Zbl 0754.00008.]
The main result of this paper is: \[ \text{(i)}\quad U{\buildrel{\text{(d)}}\over =}\theta,\qquad\text{(ii)}\quad V{\buildrel{\text{(d)}} \over =}\zeta+T_ *, \] where \(U,\theta,V\) and \(T_ *\) are Brownian stopping times defined by \[ \theta=\inf\{t\geq 0;\;\max B(u)_{0\leq u\leq t}-\min B(u)_{0\leq u\leq t}\geq 2\}, \]
\[ U=\inf\{t\geq 0;| B(t)|+\textstyle{{1\over 2}}L_ t\geq 2\},\;V=\inf\{t\geq 0; B^ +(t)+\textstyle{{1\over 2}}L_ t\geq 2\},\;T_ *=\inf\{t\geq 0;| B(t)|\geq 1\}, \] \((B(t);t\geq0)\) is a one-dimensional Brownian motion, started at 0, \((L_ t;t\geq0)\) its local time at \(0,\zeta\) is a stable r.v. with parameter \({1\over 2}\), independent of \(B\). The two identities (i) and (ii) are firstly established by a direct calculation of Laplace transforms. A probabilistic proof of (i) is given by the existence of a map which transforms \((B(t);0\leq t\leq U)\) in \((B(t);0\leq t\leq\theta)\) and preserves the law of the first process; by the same way, the author proves (ii).
Reviewer: P.Vallois (Paris)

MSC:
60J65 Brownian motion
60G17 Sample path properties
60G40 Stopping times; optimal stopping problems; gambling theory
60J55 Local time and additive functionals
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