On the passage times of self-similar Gaussian processes on curved boundaries

Abstract

Let $T_{c,\beta}$ denote the smallest $t\ge1$ that a continuous, self-similar Gaussian process with self-similarity index $\alpha>0$ moves at least $\pm c t^\beta$ units. We prove that: (i) If $\beta>\alpha$, then $T_{c,\beta}=\infty$ with positive probability; (ii) If $\beta<\alpha$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then $T_{c,\beta}$ has moments of all order; and (iii) If $\beta=\alpha$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function $\lambda:(0\,,\infty)\to(0\,,\infty)$ such that $\mathrm{E}(T_{c,\beta}^\mu)$ is finite when $0<\mu<\lambda(c)$ and infinite when $\mu>\lambda(c)$. Together these results extend a celebrated theorem of Breiman (1967) and Shepp (1967) for passage times of a Brownian motion on the critical square-root boundary. We briefly discuss two examples: One about fractional Brownian motion, and another about a family of linear stochastic partial differential equations.