Boundary crossing probabilities for stationary Gaussian processes and Brownian motion

Abstract:

Let $X(t)$ be a stationary Gaussian process, $f(t)$ a continuous function, and $T$ a finite or infinite interval. This paper develops asymptotic estimates for $P(X(t) \geqslant f(t)$, some $t \in T$ when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if $W(t)$ is Brownian motion, $f$ is continuously differentiable, and $T = [0,T]$ our estimate for $P(W(t) \geqslant f(t)$, some $t \in T)$ is \[ \Lambda = \int _0^T {{{(2t)}^{ - 1}}(f(t)/{t^{1/2}})\phi (f(t)/{t^{1/2}})} dt + {I_{\{ (f(t)/{t^{1/2}})’{|_{t = T}} < 0\} }}{\Phi ^ \ast }(f(T)/{T^{1/2}})\] provided $\Lambda$ is small. Here $\phi$ is the standard normal density and ${\Phi ^ \ast }$ is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.