Joint distributions of partial and global maxima of a Brownian Bridge

TL;DR

This paper derives exact joint and marginal distributions for the partial and global maxima of a Brownian Bridge, analyzing their correlations and moments to understand their statistical relationship over time.

Contribution

It provides the first exact analytical expressions for the joint distribution of partial and global maxima of a Brownian Bridge, including moments and correlation measures.

Findings

Exact formulas for joint distribution P(m,M)

Distribution P(m) of the partial maximum

Distribution Π(G) of the gap G = M - m

Abstract

We analyze the joint distributions and temporal correlations between the partial maximum $m$ and the global maximum $M$ achieved by a Brownian Bridge on the subinterval $[0,t_1]$ and on the entire interval $[0,t]$, respectively. We determine three probability distribution functions: The joint distribution $P(m,M)$ of both maxima; the distribution $P(m)$ of the partial maximum; and the distribution $\Pi(G)$ of the gap between the maxima, $G = M-m$. We present exact results for the moments of these distributions and quantify the temporal correlations between $m$ and $M$ by calculating the Pearson correlation coefficient.