Boundary noncrossings of additive Wiener fields∗

Abstract

Let {Wi(t), t ∈ ℝ+}, i = 1, 2, be two Wiener processes, and let W3 = {W3(t),t ∈ ℝ+2} be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability Pf = P{W1(t1) + W2(t2) + W3(t) + f(t) ≤ u(t),t ∈ ℝ+2}, where f, u : ℝ+2→ ℝ are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, Pγf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \), where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W1(t1) + W2(t2) + W3(t). It turns out that our approach is also applicable for the additive Brownian pillow.