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

TL;DR

This paper investigates the passage times of self-similar Gaussian processes crossing curved boundaries, extending classical results for Brownian motion to broader classes of Gaussian processes with applications to fractional Brownian motion and SPDEs.

Contribution

It provides new conditions and results on the finiteness and moments of passage times for self-similar Gaussian processes crossing curved boundaries, generalizing classical Brownian motion theorems.

Findings

If boundary grows faster than process, passage time is infinite with positive probability.

For slower boundary growth, passage times have moments of all orders.

At critical boundary growth, moments depend on a decreasing function of boundary parameter.

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.