Electronic Communications in Probability Constructions of a Brownian Path with a given Minimum

We construct a Brownian path conditioned on its minimum value over a xed 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 = (B t) 0t1 with B 0 = 0, one well known construction of a Brownian bridge of length 1 from 0 to x, denoted B br;x , is the following: B br;x u := B u ? uB 1 + ux (0 u 1): (1) Then a Brownian meander of length 1 starting at 0 and conditioned to end at r 0, de