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The conditional expectation and conditional variance of Brownian motion isconsidered given the argmax, B(t|argmax), as well as those with additionalinformation: 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 theexpectation and variance of a Brownian meander in time. By splicing togethertwo Brownian meanders, the mean and variance of the constrained process arecalculated. Computational results displaying both the expectation and variancein time are presented. Comparison of the simulation with theoretical values areshown when the close and argmax are given.

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