A note on multivariate Gauss-Hermite quadrature

The nodes xi and weights wi are uniquely determined by the choice of the domain D and the weighting kernel ψ(x). In fact, one may go as far as to say that the choice of the domain and the kernel defines a quadrature. In particular, the location of the nodes xi are given by the roots of the polynomial of order m in the sequence of orthonormal polynomials {πj} generated by the metric 〈πj|πk〉 := ∫ D πj(x)πk(x)ψ(x) dx = δjk, and the weights wi can be computed from a linear system once the roots are known. The mathematics of quadrature methods is well understood and described in most textbooks on numerical analysis [PTVF92]. In the case of the integration domain to be the entire real axis, and the integration kernel given by the density of a standard normal distribution, the associate quadrature scheme is known under the name GaussHermite since the involved orthogonal polynomials turn out to be Hermite polynomials. Gauss-Hermite quadrature is of fundamental importance in many areas of applied mathematics that uses statistical representations, e.g. financial mathematics and actuarial sciences. Reliable routines for the calculation of the roots and weights are readily available [PTVF92] and