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A short proof of an identity for a Brownian bridge due to Donati-Martin, Matsumoto and Yor. (English) Zbl 1110.60077

Summary: Let \((W_t)_{0\leq t\leq 1}\) be a Brownian bridge. Then, as shown by C. Donati-Martin, H. Matsumoto and M. Yor [in: Mathematical finance – Bachelier congress 2000, 221–243 (2002; Zbl 1030.91029)] the following identity holds: \[ \mathbb{E}\left[ \left(\int^t_0e^{\alpha W_t}dt\right)^{-1}\right]= 1. \] We give an elegant direct, proof of this result, based on an identification between a Brownian bridge and a Brownian excursion due to W. Vervaat [Ann. Probab. 7, 143–149 (1979; Zbl 0392.60058)] and Ph. Biane [Ann. Inst. Henri Poincaré, Probab. Stat. 22, 1–7 (1986; Zbl 0596.60079)].

MSC:

60J65 Brownian motion
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[1] Biane, Ph., Relations entre pont et excursion du mouvement Brownian réel, Ann. Inst. H. Poincaré B, 22, 1-7 (1986) · Zbl 0596.60079
[2] Bismut, J. M., Last exit decompositions and regularity at the boundary of transition probability, Zeit. für Wahr., 69, 65-99 (1985) · Zbl 0551.60077
[3] Chaumont, L.; Hobson, D. G.; Yor, M., Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes, Sém. Prob., XXXV, 334-347 (2001) · Zbl 0982.60020
[4] Donati-Martin, C., Matsumoto, H., Yor, M., 1999. The law of geometric Brownian motion and its integral, revisited: application to conditional moments. Preprint.; Donati-Martin, C., Matsumoto, H., Yor, M., 1999. The law of geometric Brownian motion and its integral, revisited: application to conditional moments. Preprint. · Zbl 1030.91029
[5] Donati-Martin, C.; Matsumoto, H.; Yor, M., On a striking identity about the exponential functional of the Brownian bridge, Period. Math. Hungar., 41, 103-119 (2000) · Zbl 1062.60080
[6] Lyasoff, A., 2005. On the distribution law of the integral of geometric Brownian motion. Preprint, Boston University, \(2005 \langle;\) http://andrew.lyasoff.com/igbm/paper_igbm_AL.pdf \(\rangle;\); Lyasoff, A., 2005. On the distribution law of the integral of geometric Brownian motion. Preprint, Boston University, \(2005 \langle;\) http://andrew.lyasoff.com/igbm/paper_igbm_AL.pdf \(\rangle;\)
[7] Vervaat, W., A relation between Brownian bridge and Brownian excursion, Ann. Probab., 7, 143-149 (1979) · Zbl 0392.60058
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