The determinant of the iterated Malliavin matrix and the density of a pair of multiple integrals (Q516127)

The authors consider a two-dimensional random vector whose components are multiple stochastic integrals of the same order \(n\). It is shown that such a random vector either admits a density with respect to the Lebesgue measure, or its components are proportional. As an auxiliary result, it is shown that in the case when the determinant of an iterated Malliavin matrix of a pair of multiple integrals vanishes, then the determinant of any other iterated Malliavin matrix will vanish, too. Also, a representation of the determinant of the \(k\)-th iterated Malliavin matrix is obtained, and an upper bound 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 determinants of its iterated Malliavin matrices is established, in order to prove the ``proportionality result''.