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

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 leads to completely explicit formulas for the Laplace transform.