The joint distributions of running maximum of a Slepian processes

Consider the Slepian process $S$ defined by $S(t)=B(t+1)-B(t),t\in [0,1]$ with $B(t),t\in \R$ a standard Brownian motion.In this contribution we analyze the joint distribution between the maximum $m_{s}=\max_{0\leq u\leq s}S(u)$ certain and the maximum $M_t=\max_{0\leq u\leq t}S(u)$ for $0< s < t$ fixed. Explicit integral expression are obtained for the distribution function of the partial maximum $m_{s}$ and the joint distribution function between $m_{s}$ and $M_t$. We also use our results to determine the moments of $m_{s}$.