Sunset over Brownistan (Q1185783)

Let \(B_ t\) be a standard Brownian motion starting at zero and \(X_{t,a}=B_ t-at\) be a Brownian motion with the downward drift of rate \(a>0\). The author investigates \(Z_ a=\max_{t\geq 0} X_{t,a}\) and \(D_ a=\sup\{t: B_ t-at=Z_ a\}\) as the stochastic processes indexed by \(a\). Using simple geometric observations he discovers the relation between \(Z_ a\) and the concave majorant of \(B_ t\) and establishes the characterization of \(Z_ a\) and \(D_ a\) in terms of Poisson random mmeasure. This gives the opportunity to obtain some distributional results for \(Z_ a\) and \(D_ a\) which are used in queueing theory.