Mean and Variance of Brownian Motion with Given Final Value, Maximum and ArgMax: Extended Version

Abstract

The conditional expectation and conditional variance of Brownian motion is considered given the argmax, B(t|argmax), as well as those with additional information: B(t|close, argmax), B(t|max, argmax), B(t|close, max, argmax) where the close is the final value: B(t=1)=c and t in [0,1]. We compute the expectation and variance of a Brownian meander in time. By splicing together two Brownian meanders, the mean and variance of the constrained process are calculated. Computational results displaying both the expectation and variance in time are presented. Comparison of the simulation with theoretical values are shown when the close and argmax are given.