Boundary Non-crossings of Brownian Pillow

Let B ₀(s,t) be a Brownian pillow with continuous sample paths, and let h,u:0,1]²→? be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability

$\psi(u;h):=\mathbf{P}\big\{B_{0}(s,t)+h(s,t)\leq u(s,t),\forall s,t\in0,1]\big\}.$

Further we investigate the asymptotic behaviour of ψ(u;γ h) with γ tending to ∞ and solve a related minimisation problem.