Joint distributions of partial and global maxima of a Brownian bridge

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 Π ( 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.

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