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

Abstract

Let $T_{c,\beta }$ denote the smallest $t\ge 1$ 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.