Boundary Non-Crossings of Additive Wiener Fields

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 $γf>0$ asymptotically when $γ\to \infty$ we have that $\ln P_{γf}$ is the same as $\ln P_{γ\underline{f}}$ 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.

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