Matsumoto, Hiroyuki; Yor, Marc A version of Pitman’s \(2M-X\) theorem for geometric Brownian motions. (English. Abridged French version) Zbl 0936.60076 C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 11, 1067-1074 (1999). The authors consider the geometric Brownian motion with parameter \(\mu\), i.e., the exponential of one-dimensional Brownian motion with drift \(\mu\). They show that this process divided by its quadratic variation process, is a diffusion process. Then, taking logarithms and an appropriate scaling limit, they recover the Rogers-Pitman extension to Brownian motion with drift of Pitman’s representation theorem for the three-dimensional Bessel process, see J. W. Pitman [Adv. Appl. Probab. 7, 511-526 (1975; Zbl 0332.60055)], and L. C. G. Rogers and J. W. Pitman [Ann. Probab. 9, 573-582 (1981; Zbl 0466.60070)]. Time-reversal and generalized inverse Gaussian distributions play crucial parts in the proofs. Reviewer: Marius Iosifescu (Bucureşti) Cited in 1 ReviewCited in 14 Documents MSC: 60J65 Brownian motion 60J60 Diffusion processes Keywords:geometric Brownian motion; Rogers-Pitman extension; three-dimensional Bessel process; generalized inverse Gaussian distributions Citations:Zbl 0332.60055; Zbl 0466.60070 PDFBibTeX XMLCite \textit{H. Matsumoto} and \textit{M. Yor}, C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 11, 1067--1074 (1999; Zbl 0936.60076) Full Text: DOI