Constructions of a Brownian Path with a given Minimum

We construct a Brownian path conditioned on its minimum value over a fixed time interval by a simple transformation of a Brownian bridge. DOI: 10.1214/ECP.v4-1003 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-79487 Published Version Originally published at: Bertoin, Jean; Pitman, Jim; de Chavez, Juan Ruiz (1999). Constructions of a Brownian path with a given minimum. Electronic Communications in Probability, 4:31-37. DOI: 10.1214/ECP.v4-1003 Elect. Comm. in Probab. 4 (1999) 31–37 ELECTRONIC COMMUNICATIONS in PROBABILITY CONSTRUCTIONS OF A BROWNIAN PATH WITH A GIVEN MINIMUM JEAN BERTOIN Laboratoire de Probabilités, tour 56, Université Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris Cedex 05, France email: [email protected] JIM PITMAN Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA email: [email protected] JUAN RUIZ DE CHAVEZ Depto. de Matemáticas UAM-I, Apdo. Postal 55-534, C.P. 09340 México D.F., México email: [email protected] submitted February 25, 1999; accepted July 9, 1999 AMS subject classification: 60J65 Conditioned Brownian motion, path transformations Abstract We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations of a Brownian bridge.We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations of a Brownian bridge. Path transformations have proved useful in the study of Brownian motion and related processes, by providing simple constructions of various conditioned processes such as Brownian bridge, meander and excursion, starting from an unconditioned Brownian motion. As well as providing insight into the structure of these conditioned processes, path constructions assist in the computation of various conditional laws of Brownian functionals, and in the simulation of conditioned processes. Starting from a standard one-dimensional Brownian motion B = (Bt)0≤t≤1 with B0 = 0, one well known construction of a Brownian bridge of length 1 from 0 to x, denoted B, is the following: B u := Bu − uB1 + ux (0 ≤ u ≤ 1). (1) Then a Brownian meander of length 1 starting at 0 and conditioned to end at r ≥ 0, denoted B, can be constructed from three independent copies (B i,u )0≤u≤1, i = 1, 2, 3 of the