Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Some changes of probabilities related to a geometric Brownian motion version of Pitman's 2M - X theorem.

Auteur(s):

Code(s) de Classification MSC:

• 60G44 Martingales with continuous parameter
Résumé: Rogers-Pitman have shown that the sum of the absolute value of $B^{\mu}$, Brownian motion with constant drift $\mu$, and its local time $L^{\mu}$ is a diffusion $R^{\mu}$. We exploit the intertwining relation between $B^{\mu}$ and $R^{\mu}$ to show that the same addition operation performed on a one-parameter family of diffusions $\{X^{(\alpha, \mu)}\}_{\alpha \in {\mathbb R}_+}$ yields the same diffusion $R^{\mu}$. Recently we obtained an exponential analogue of the Rogers-Pitman result. Here we exploit again the corresponding intertwining relationship to yield a one-parameter family extension of our result.