Various methods to derive new formulas for the Laplace transforms of some
quadratic forms of Gaussian sequences are discussed. In the general setting, an
approach based on the resolution of an appropriate auxiliary filtering problem is
developed; it leads to a formula in terms of the solutions of Volterra-type
recursions describing characteristics of the corresponding optimal filter. In the
case of Gauss-Markov sequences, where the previous equations reduce to
ordinary forward recursive equations, an alternative approach prices another
formula; it involves the solution of a backward recursive equation. Comparing
the different formulas for the Laplace transforms, various relationships between
the corresponding entries are identified. In particular, relationships between the
solutions of matched forward and backward Riccati equations are thus proved
probabilistically; they are proved again directly. In various specific cases, a
further analysis of the concerned equations lead to completely explicit formulas
for the Laplace transform.
Recursive equations for arbitrary scattering processes
