Some Changes of Probabilities Related to a Geometric Brownian Motion Version of Pitman's 2M-X Theorem
Hiroyuki Matsumoto, Nagoya University
Marc Yor, Universite Pierre et Marie Curie
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
${X^{(alpha,mu)}}_{alphain{mathbf R}_+}$ 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.
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