Sequential Gaussian Markov Integrals

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...

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...

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...

Gaussian Approximations of Multiple Integrals

Fix k ≥ 1, and let I(l), l ≥ 1, be a sequence of k-dimensional vectors of multiple Wiener-Itô integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l → +∞ , the law of I(l) is asymptotically close (for example, in the sense of Prokhorov’s distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as...