New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences

Various methods to derive new formulas for the Laplace transforms of somequadratic forms of Gaussian sequences are discussed. In the general setting, anapproach based on the resolution of an appropriate auxiliary filtering problem isdeveloped; it leads to a formula in terms of the solutions of Volterra-typerecursions describing characteristics of the corresponding optimal filter. In thecase of Gauss-Markov sequences, where the previous equations reduce toordinary forward recursive equations, an alternative approach prices anotherformula; it involves the solution of a backward recursive equation. Comparingthe different formulas for the Laplace transforms, various relationships betweenthe corresponding entries are identified. In particular, relationships between thesolutions of matched forward and backward Riccati equations are thus provedprobabilistically; they are proved again directly. In various specific cases, afurther analysis of the concerned equations lead to completely explicit formulasfor the Laplace transform