Abstract
The operation of random Brownian scaling is introduced. Applied to the random intervals (0, g), (g, d), (g, 1), where g is the last Brownian zero before time 1, respectively, d is the first Brownian zero after time 1, it is shown that the corresponding Brownian scaled processes are respectively the Brownian bridge, the BES(3) bridge, and the Brownian meander. Independence properties of the Brownian meander allow to study Azéma’s remarkable martingale, which enjoys the chaos representation property, as shown by Emery.
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Yen, JY., Yor, M. (2013). A Simple Path Decomposition of Brownian Motion Around Time t = 1. 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_7
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DOI: https://doi.org/10.1007/978-3-319-01270-4_7
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