First Passage Time for a Particular Gaussian Process

Abstract

We find an explicit formula for the first passage probability, $Q_a(T | x) = P_r(S(t) < a, 0 \leqq t \leqq T | S(0) = x)$, for all $T > 0$, where $S$ is the Gaussian process with mean zero and covariance $ES(\tau)S(t) = \max (1 - |t - \tau|, 0)$. Previously, $Q_a(T\mid x)$ was known only for $T \leqq 1$. In particular for $T = n$ an integer and $-\infty < x < a < \infty$, $Q_a(T \mid x) = \frac{1}{\varphi(x)} \int_D \cdots \int \det \varphi(y_i - y_{j+1} + a) dy_2 \cdots dy_{n+1},$ where the integral is an $n$-fold integral on $y_2, \cdots, y_{n+1}$ over the region $D$ given by $D = \{a - x < y_2 < y_1 < \cdots < y_{n+1}\}$ and the determinant is of size $(n + 1) \times (n + 1), 0 < i, j \leqq n$, with $y_0 \equiv 0, y_1 \equiv a - x$.