On rice's formula for stationary multivariate piecewise smooth processes

Abstract

Let X = {Xt: t ≥ 0} be a stationary piecewise continuous ℝd -valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ ℝd 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. © Applied Probability Trust 2012.