Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

Abstract

We study boundary non-crossing probabilitiesfor continuous centered Gaussian process X indexed by some arbitrary compact separable metric space T. We obtain both upper and lower bounds for P f,u . The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case P cf,u , c → +∞, which are two-term approximations (up to o(c)). The asymptotics are formulated in terms of the solutionf to the constrained optimization problem h HX → min, h ∈ H X , h ≥ f in the reproducing kernel Hilbert space H X of X. Several applications of the results are further presented.2010 Mathematics Subject Classification. 60G15; 60G70; 60F10.