Conditionally Gaussian stochastic integrals

Conditionally Gaussian stochastic integrals

We derive conditional Gaussian type identities of the form E [ exp ( i ∫ T 0 utdBt ) ∣∣∣∣ ∫ T 0 |ut|dt ] = exp ( − 2 ∫ T 0 |ut|dt ) , for Brownian stochastic integrals, under conditions on the process (ut)t∈[0,T ] specified using the Malliavin calculus. This applies in particular to the quadratic Brownian integral ∫ t 0 ABsdBs under the matrix condition A †A2 = 0, using a characterization of Yo...

Gaussian limits for vector-valued multiple stochastic integrals

We establish necessary and sufficient conditions for a sequence of d-dimensional vectors of multiple stochastic integrals Fd = ` F k 1 , ..., F k d ́ , k ≥ 1, to converge in distribution to a d-dimensional Gaussian vector Nd = (N1, ..., Nd). In particular, we show that if the covariance structure of F k d converges to that of Nd, then componentwise convergence implies joint convergence. These re...

Gaussian integrals

II.G Gaussian Integrals

It can be reduced to a product of N one dimensional integrals by diagonalizing the matrix K ≡ Ki,j . Since we need only consider symmetric matrices (Ki,j = Kj,i), the eigenvalues are real, and the eigenvectors can be made orthonormal. Let us denote the eigenvectors and eigenvalues of K by q̂ and Kq respectively, i.e. Kq̂ = Kqq̂. The vectors {q̂} form a new coordinate basis in the original N dimensi...

II.G Gaussian Integrals

It can be reduced to a product of N one dimensional integrals by diagonalizing the matrix K ≡ Ki,j . Since we need only consider symmetric matrices (Ki,j = Kj,i), the eigenvalues are real, and the eigenvectors can be made orthonormal. Let us denote the eigenvectors and eigenvalues of K by q̂ and Kq respectively, i.e. Kq̂ = Kqq̂. The vectors {q̂} form a new coordinate basis in the original N dimensi...