The determinant of the iterated Malliavin matrix and the density of a couple of multiple integrals

TL;DR

This paper provides an estimate for the determinant of the covariance matrix of a two-dimensional vector of multiple stochastic integrals, linking it to the iterated Malliavin matrices, and characterizes absolute continuity.

Contribution

It introduces a new estimate relating the covariance determinant to iterated Malliavin matrices and characterizes when the vector's distribution is absolutely continuous.

Findings

The covariance determinant can be estimated using iterated Malliavin matrices.

The vector is absolutely continuous if and only if its components are proportional.

Provides a criterion for absolute continuity based on the structure of the components.

Abstract

The aim of this paper is to show an estimate for the determinant of the covariance of a two-dimensional vector of multiple stochastic integrals of the same order in terms of a linear combination of the expectation of the determinant of its iterated Malliavin matrices. As an application we show that the vector is absolutely continuous if and only if its components are proportional.