In this contribution we derive an explicit formula for the boundary non-crossing probabilities for Slepian processes associated with the piecewise linear boundary function. This formula is used to develop an approximation formula to the boundary non-crossing probabilities for general continuous boundaries. The formulas we developed are easy to implement in calculation the boundary non-crossing probabilities.

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Consider the Slepian process S defined by S(t) = B(t + 1) − B(t), t ∈ [0, 1] with B(t), t ∈ R a standard Brownian motion. In this contribution we analyze the joint distribution between the maximum m s = max 0≤u≤s S(u) certain and the maximum M t = max 0≤u≤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 .