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

TL;DR

This paper derives the conditional expectation and variance of Brownian motion given its argmax, final value, and maximum, providing theoretical formulas and computational results for constrained Brownian processes.

Contribution

It extends the analysis of Brownian motion by deriving explicit formulas for conditional moments given multiple path-dependent conditions, including argmax and final value.

Findings

Explicit formulas for mean and variance of Brownian motion with given argmax and final value.

Comparison of simulation results with theoretical values confirms accuracy.

Method for splicing Brownian meanders to analyze constrained processes.

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.