Numerical calculation of the level crossing rate of second order stochastic Volterra systems

A new method for calculating the mean level crossing rate of a stochastic process represented as a second order stochastic Volterra series is discussed. Its derivation is carried through in three steps. First, the expression for the mean crossing rate is rewritten in terms of a joint characteristic function. Secondly, it is shown that a closed form expression for this joint characteristic function can be derived. Thirdly, it is then demonstrated how the method of steepest descent can be applied to the numerical calculation of the mean crossing rate. It is shown by an example that the numerical accuracy of this method is apparently very high.