M. Viot

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 Voterra 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 provides another formula; it involves the solution of a backward recursive equation. Comparing the different formulas for the Laplace transform- s, 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.

On the Linear-Exponential Filtering Problem for General Gaussian Processes

The explicit solution of the filtering problem with exponential criteria for a general Gaussian signal is obtained through an approach which is based on a conditional Cameron-Martin type formula. This key formula is derived for conditional expectations of exponentials of some quadratic functionals of a general continuous Gaussian process. The formula involves conditional expectations and conditional covariances in some auxiliary optimal risk-neutral filtering problem.

Risk sensitive and LEG filtering problems are not equivalent

Filtering problems with general exponential quadratic criteria are investigated for Gauss-Markov processes. In this setting, the linear exponential Gaussian and risk sensitive filtering problems are solved and it is shown that they may have different solutions.

On the linear-exponential filtering problem for general Gaussian processes

The explicit solution of the filtering problem with exponential criteria for a general Gaussian signal is obtained through an approach which is based on a conditional Cameron-Martin type formula. This key formula is derived for conditional expectations of exponentials of some quadratic functionals of a general continuous Gaussian process. The formula involves conditional expectations and conditional covariances in some auxiliary optimal risk-neutral filtering problem.