Abstract
This chapter is devoted to the discussion of two famous theorems: the first one, due to Lévy, asserts that, if one subtracts Brownian motion from its one-sided supremum, the obtained process is distributed as the absolute value of Brownian motion; the second one, due to Pitman, asserts that if one subtracts Brownian motion from twice its one sided supremum, the obtained process is distributed as a BES(3) process. Extensions of these theorems to Brownian motion with drift are shown. The Azéma–Yor explicit solution to Skorokhod’s embedding problem is shown; it involves a first hitting time by Brownian motion and its one-sided supremum.
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Notes
- 1.
It is even more direct to see this identity as a consequence of the switching identity:
$$\displaystyle{E[F(B_{u}; u \leq \tau _{1})\vert \tau _{1} = t] = E[F(B_{u}; u \leq t)\vert B_{t} = 0,L_{t} = 1].}$$
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Yen, JY., Yor, M. (2013). Lévy’s Representation of Reflecting BM and Pitman’s Representation of BES(3). 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_3
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DOI: https://doi.org/10.1007/978-3-319-01270-4_3
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