Optimal linear filtering of general multidimensional Gaussian processes - Application to Laplace transforms of quadratic functionals

The optimal filtering problem for multidimensional continuous possibly non-Markovian, Gaussian processes, observed through a linear channel driven by a Brownian motion, is revisited. Explicit Volterra type filtering equations involving the covariance function of the filtered process are deriv-ed both for the conditional mean and for the covariance of the filtering error. The solution of the filtering problem is applied to obtain a Cameron-Martin type formula for Laplace transforms of a quadratic func-tional of the process. Particular cases for which the results can be further elaborated are investigated