×

A version of Pitman’s \(2M-X\) theorem for geometric Brownian motions. (English. Abridged French version) Zbl 0936.60076

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.

MSC:

60J65 Brownian motion
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI