The rate of crossings of a quadratic form of an n-dimensional stationary Gaussian process

Consider a quadratic form of a vector valued differentiable stationary Gaussian process. The crossing intensity of a fixed level depends on the joint correlation structure of the process and its derivative, but no simple exact form is known for the general case. We give the first and second terms in an asymptotic expansion, which is valid as the level tends to infinity, and show how to find higher order terms. As a by-product of the proof we see that the crossing intensity can be written as an integral which, even if it cannot be solved by a simple formula, is easily evaluated by Monte Carlo integration. We give a simulation scheme to describe the steps in this procedure