Boundary non-crossings of Brownian pillow (Q966499)

Let \(B_0(s,t),\;s,t\in [0,1]\) be a Brownian pillow with continuous sample paths having covariance function \(K\) which is equal to the product of two covariance functions of a Brownian bridge: \[ K((s_1,t_1),(s_2,t_2))=K_1(s_1,t_1)K_2(s_2,t_2),\;\;\;s_i,t_i\in [0,1],i=1,2; \] \(K_i(s,t)=\min(s,t)-st,\;i=1,2.\) Then the concern of this article is the boundary non-crossing probability \[ \psi(u;h):=P\{B_0(s,t)+h(s,t)\leq u(s,t),\forall s,t\in [0,1]\} \] with a trend function \(h\) and a measurable boundary function \(u\). When considering Brownian bridge or a Brownian motion, the corresponding non-crossing probability can be explicitly calculated if \(h\) and \(u\) are polygonal lines. Such formulae are not available in author's setup of the multi-parameter processes. That's why the novel results presented here are: 1) upper and lower bounds for \(\psi(u;h),\) 2) a large deviation type result for the boundary non-crossing probability \(\psi(u;\gamma h)\) with \(\gamma \to \infty,\) and 3) a related minimisation problem.