Boundary Non-Crossings of Additive Wiener Fields

Abstract

Let $W_i=\{W_i(t), t\in \mathbb{R}_+\}, i=1,2$ be two Wiener processes and $W_3=\{W_3(\mathbf{t}), \mathbf{t}\in \mathbb{R}_+^2\}$ be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary non-crossing probability $$P_f=P\{W_1(t_1)+W_2(t_2)+W_3(\mathbf{t})+h(\mathbf{t})\leq u(\mathbf{t}), \mathbf{t}\in\mathbb{R}_+^2\},$$ where $h, u: \mathbb{R}_+^2\rightarrow \mathbb{R}_+$ are two measurable functions. We show further that for large trend functions $\gamma f>0$ asymptotically when $\gamma \to \infty$ we have that $\ln P_{\gamma f}$ is the same as $\ln P_{\gamma \underline{f}}$ where $\underline{f}$ is the projection of $f$ on some closed convex set of the reproducing kernel Hilbert Space of $W$. It turns out that our approach is applicable also for the additive Brownian pillow.