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Zbl 0621.60081
Csáki, Endre; Földes, Antónia; Salminen, Paavo
On the joint distribution of the maximum and its location for a linear diffusion. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 23, 179-194 (1987). ISSN 0246-0203

Let X be a regular one-dimensional diffusion and let M be its maximum value before time t, and T (almost surely unique) time at which $X=M$ before t. A general formula is given for the joint distribution of $X\sb t$, M, and T, in terms of the hitting-time densities, the scale function, and the speed measure of X. \par Two proofs are given; one is probabilistic and the other is analytical. Various examples are discussed, including the diffusion representation of Brownian local time evaluted at an exponential time.
[W.S.Kendall]

MSC 2000:: *60J60 Diffusion processes
60J55 Additive functionals

Keywords: hitting-time densities; diffusion representation of Brownian local time; exponential time

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