Product formula and independence criterion for multiple Huang–Cambanis integrals

Abstract

Multiple stochastic integrals of Huang and Cambanis [1978. Stochastic and multiple Wiener integrals for Gaussian processes. Ann. Probab. 6, 585–614] with respect to a general Gaussian process X = ( X t , t ∈ T ) , whose covariance function is of bounded variation on bounded subsets of T × T , are considered. A product formula for the integrals is derived and a necessary and sufficient condition for independence of multiple Huang–Cambanis integrals is obtained. As an illustration, the results are applied to the special case of multiple integrals with respect to a persistent fractional Brownian motion.