On the density of some Wiener functionals: An application of Malliavin calculus (Q2366862)

Let \(X\) be a real random variable with a distribution function \(F(x)\). Consider a random variable \(T=\int^{T_ 2}_{T_ 1}[W(h_ t)]^ 2dH(t)\), where \(W(h)\) is a Wiener integral of a square integrable function on \([0,1]\) with respect to the Wiener process, \(h_ t(y)=\sin(tF^{-1}(y))-\text{Im}(E\exp\{itX\})\), and \(H(t)\) is a distribution function on a finite interval \([T_ 1,T_ 2]\). By using Malliavin's calculus, a representation \(T=\sum\lambda_ iG^ 2_ i\) is established, where the series of nonincreasing positive real numbers \(\lambda_ i\) converges and \(G_ i\)'s are independent standard Gaussian random variables. The above representation is used to deduce, by estimating the Fourier transform of \(T\), that \(T\) must necessarily have a smooth density belonging to the class of tempered distributions. A connection to empirical characteristic processes in statistics is also explained.