Brownian Motion Conditioned to Spend Limited Time Below a Barrier

Abstract

We condition a Brownian motion with arbitrary starting point y ∈ R on spending at most 1 time unit below 0 and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero g = g y and of its occupation time Γ = Γ y below 0 as functions of y. This generalizes Theorem 4 of [BB11], which covers the special case y = 0. Additionally, we study the behavior of the distributions of g y and Γ y , respectively, for y → ±∞.