On relation between one multiple and corresponding one-dimensional integral with applications

Abstract

For a given finite positive measure on an interval I R , we introduce a multiple stochastic integral of a Volterra kernel with respect to a product of a corresponding Gaussian orthogonal stochastic measure. Indicating that the previous defined multiple stochastic integral is in relation with a parameterized Hermite polynomial of a suitable stochastic integral, that is, of a suitable Gaussian random variable, we prove that one multiple integral can be expressed by a corresponding one-dimensional. Having in mind the obtained result, we show that a collection of the multiple integrals can be integrated exactly by a Gaussian quadrature rule. In particular, under certain conditions, a classical Gaussian quadrature rule can be used to approximate the value of one type of the multiple integral. A probabilistic interpretation is given.
Keywords: Multiple stochastic integral, Multiple integral, Gaussian quadrature rule.