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Un processus qui ressemble au pont Brownien. (A process resembling the Brownian bridge). (French) Zbl 0621.60086

Sémin. probabilités XXI, Lect. Notes Math. 1247, 270-275 (1987).
[For the entire collection see Zbl 0606.00022.]
Let \((B_ t)\) be a real Brownian motion starting at 0 and \((\ell_ t,t\geq 0)\) its local time at 0, \(\tau_ t=\inf \{u:\ell_ u>t\}\). Put \(X_ u=B_{u\tau_ 1}/\sqrt{\tau_ 1}\), \(p(u)=\) Brownian bridge, and \(\lambda =\) its local time at level 0 at time 1.
The main result asserts that, for each Borel functional \(F: C([0,1],{\mathbb{R}})\to {\mathbb{R}}_+\), one has the identity \[ E(F(X_ u,0\leq u\leq 1))=E(F(p(u),0\leq u\leq 1)\sqrt{2/\pi}/\lambda). \]
Reviewer: U.Krengel

MSC:

60J65 Brownian motion

Citations:

Zbl 0606.00022
Full Text: Numdam EuDML