The Expected Number of Level Crossings for Stationary, Harmonisable, Symmetric, Stable Processes

Abstract

Let $\{X(t), t \geq 0\}$ be a harmonisable, symmetric, $\alpha$-stable stochastic process and let $C_u(T)$ be the number of times that $X$ crosses the level $u$ during the time interval $\lbrack 0, T\rbrack$. Our main result is the precise numerical value of $C := \lim_{u \rightarrow \infty} u^\alpha EC_u(T)$. By way of examples, including an explicit evaluation of $EC_u$ for a stationary process and a combination of analytic and Monte Carlo techniques for some others, we show that the asymptotic approximation $EC_u \sim Cu^{-\alpha}$ is remarkably accurate, even for quite low values of the level $u$. This formula therefore serves, for all practical purposes, as a "Rice formula" for harmonisable stable processes, and should be as important in the applications of harmonisable stable processes as the original Rice formula was for their Gaussian counterparts. We also have upper and lower bounds for $EC_u$ that hold for all $u$ and that, unlike previous results in the area, also hold for all $\alpha$ and are of the correct order of magnitude for large $u$.