A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration

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Exponential functionals of Brownian motion, II: Some related diffusion processes (2005, background): “2) at the process level has been given in [47].”
An Analogue of Pitman’s 2M — X Theorem for Exponential Wiener Functionals Part II: The Role of the Generalized Inverse Gaussian Laws (2001, background): “In the present Part II, we show that these results may be derived as a consequence of the following extension given in [17] of Dufresne’s identity [9].”
Brownian analogues of Burke's theorem (2001, methodology): “…A−2s exp(2B ( ) s ) ds and (19) follows by noting that dA−1s =− A−2s exp(2B( )s ) ds; and hence∫ s¿t A−2s exp(2B ( ) s ) ds=A −1 t : The proof of Theorem 6(2) given in Matsumoto and Yor (2001) uses the method of enlargement; in Section 7 we will give an alternative proof using reversibility…”