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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2008

On the time to reach maximum for a variety of constrained Brownian motions

Résumé

We derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes.
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Dates et versions

hal-00257348 , version 1 (19-02-2008)

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Satya. N. Majumdar, Julien Randon-Furling, Michael J. Kearney, Marc Yor. On the time to reach maximum for a variety of constrained Brownian motions. Journal of Physics A: Mathematical and Theoretical, 2008, 41 (36), pp.365005. ⟨10.1088/1751-8113/41/36/365005⟩. ⟨hal-00257348⟩
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