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

Abstract

which is the quadratic variation process of the corresponding geometric Brownian motion (μ) = { (μ) = exp ( (μ)) ≧ 0}. These Wiener functionals play important roles in many fields; mathematical finance (see, e.g., Yor [33], [34], Geman-Yor [10], [11], Leblanc [17]), diffusion processes in random environment (Comtet-Monthus [5], Comtet-Monthus-Yor [6], Kawazu-Tanaka [16]), probabilistic studies related to Laplacians on hyperbolic spaces (Gruet [12], Ikeda-Matsumoto [13]) and so on. The reader will find more related topics and references in [37]. However, even for fixed , the joint law of ( (μ) (μ)) or, equivalently, that of ( = (0) ) due to the Cameron-Martin relationship between (μ) and is fairly complicated, although it is quite tractable (see [33]). As an example of the description of this law, we present the following conditional Laplace transform ([2], [13], [18]): [ exp ( − 2

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