On Relation Between One Multiple and a Corresponding One-Dimensional Integral With Applications

Tatjana Bajić

Yugoslav journal of operations research, Tome 28 (2018) no. 1.

Résumé

For a given finite positive measure on an interval $I \subseteq \R$, a multiple stochastic integral of a Volterra kernel with respect to a product of a corresponding Gaussian orthogonal stochastic measure is introduced. The Volterra kernel is taken such that the multiple stochastic integral is a multiple iterated stochastic integral related to a parameterized Hermite polynomial, where parameter depends on Gaussian distribution of an underlying one-dimensional stochastic integral. Considering that there exists a connection between stochastic and deterministic integrals, we expose some properties of parameterized Hermite polynomials of Gaussian random variable in order to prove that one multiple integral can be expressed by a corresponding one-dimensional integral. Having in mind the obtained result, we show that a system of multiple integrals, as well as a collection of conditional expectations can be calculated exactly by generalized Gaussian quadrature rule.

Détail

Mots-clés : Multiple Stochastic Integral, Multiple Integral, Gaussian Quadrature Rule