Abstract
Starting with some simple relationship between the law of a transient Bessel process, Brownian scaled at a last passage time γ, and the law of the Bessel process itself, the law of γ for a general transient diffusion in (0, ∞) is expressed in terms of the density of its semi-group, considered with respect to time. Conditioning the past of this diffusion with respect to γ = t, is shown to yield the law of the corresponding diffusion bridge of duration t. As an example, the law of the first hit of 0 by an Ornstein–Uhlenbeck process is obtained.
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Yen, JY., Yor, M. (2013). The Laws of, and Conditioning with Respect to, Last Passage Times. In: Local Times and Excursion Theory for Brownian Motion. Lecture Notes in Mathematics, vol 2088. Springer, Cham. https://doi.org/10.1007/978-3-319-01270-4_8
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DOI: https://doi.org/10.1007/978-3-319-01270-4_8
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