Exponential functionals of Brownian motion. I: Probability laws at fixed time.

Matsumoto, Hiroyuki; Yor, Marc

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Displaying similar documents to “Exponential functionals of Brownian motion. I: Probability laws at fixed time.”

Exponential functionals of Brownian motion. II: Some related diffusion processes.

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As in preceding papers in which we studied the limits of penalized 1-dimensional Wiener measures with certain functionals Γ, we obtain here the existence of the limit, as → ∞, of -dimensional Wiener measures penalized by a function of the maximum up to time of the Brownian winding process (for ), or in 2 dimensions for Brownian motion prevented to exit a cone before time . Various extensions of these multidimensional penalisations are studied, and the limit laws...

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We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_{t}^{-} : = \int_{0}^{t} 1_{X_{s} < 0} d s, t \geq 0)$ . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, (2006) 171–246]).

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