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Rogers-Pitman have shown that the sum of the absolute value of B(), Brownian motion with constant drift ,a nd its local timeL() is a diusion R(). We exploit the intertwining relation between B() and R() to show that the same addition operation performed on a one-parameter family of diusions fX(;)g2R+ yields the same diusion R(). 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.

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