On Rice's formula for stationary multivariate piecewise smooth processes

Abstract

Let X = {X_t: t ≥ 0} be a stationary piecewise continuous R^d-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ 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 X₀. Our result is illustrated by examples relating to queueing networks and stress release network models.