Abstract
Due to the scaling property of Brownian motion, it is often of interest, in order to deal with Brownian functionals up to fixed times, to consider the integral over time t of the Wiener measure restricted to the time interval (0, t). This integral is shown to be expressible in terms of both the inverse local time integral of Wiener measure and the level integral of Wiener measure up to first hit of 0 by Brownian motion, or last passage time at a level by the BES(3) process. These relations shall play a key role in our derivation of the Feynman–Kac formula in the next chapter.
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Yen, JY., Yor, M. (2013). Integral Representations Relating W and n. 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_9
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DOI: https://doi.org/10.1007/978-3-319-01270-4_9
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