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Inverse local times, positive sojourns, and maxima for Brownian motion. (English) Zbl 0665.60072

Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 233-247 (1988).
[For the entire collection see Zbl 0649.00017.]
Let B(t) be a standard Brownian motion starting at 0 and let \(\ell (t,x)\) be its local time. Denote \(\ell_ 0(t)=\ell (t,0)\) and \(T(\alpha)=\ell_ 0^{-1}(2\alpha)\). This article deals with the joint distribution of the three random variables \[ T(\alpha),\quad S^+(\alpha)=\int^{T(\alpha)}_{0}I_{(0,\infty)}(B(s))ds\quad and\quad M^+=\max_{t\leq T(\alpha)}B(t). \] As a main step in solving this problem, the author obtained the Laplace transform \(E(\exp (-\lambda S^+)| M^+=m)\) and investigated the inversion of this transform.
Reviewer: N.M.Zinchenko

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion

Citations:

Zbl 0649.00017