Some Changes of Probabilities Related to a Geometric Brownian Motion Version of Pitman's $2M-X$ Theorem

Abstract

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 ${X^{(\alpha,\mu)}}_{\alpha\in{\mathbf R}_+}$ 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.