On Rice's formula for stationary multivariate piecewise smooth processes

Abstract

Let $X=\{X_t:t\ge 0\}$ be a stationary piecewise continuous ${\mathbb{R}}^d$-valued process that moves between jumps along the integral curves of a given continuous vector field, and let $S\subset {\mathbb{R}}^d$ be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of $S$ by $X$ to the distribution of $X_0$. Our result is illustrated by examples relating to queueing networks and stress release network models.