Boundary noncrossings of additive Wiener fields∗
Hashorva, Enkelejd ; Mishura, Yuliya
In: Lithuanian Mathematical Journal, 2014, vol. 54, no. 3, p. 277-289
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- Let {W i (t), t ∈ ℝ+}, i = 1, 2, be two Wiener processes, and let W 3 = {W 3(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 P f = P{W 1(t 1) + W 2(t 2) + W 3(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 → ∞, P γf is equivalent to P γ f ¯ $$ {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} $$ , where f ¯ $$ \underset{\bar{\mkern6mu}}{f} $$ is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W 1(t 1) + W 2(t 2) + W 3(t). It turns out that our approach is also applicable for the additive Brownian pillow.