Optimal linear filtering of general multidimensional Gaussian processes and its 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 derived 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 functional of the process. Particular cases for which the results can be further elaborated are investigated.