Calcul stochastique appliqué aux problèmes de détection des signaux aléatoires
Signal detection is one of the basic problems in statistical communication theory, and has many applications to contemporary technology, whether in engineering, medical science, or the environment. The most difficult problems are those involving random signals, and it is these types of signals that are found in applications to complex systems (the ocean, the atmosphere, the ecosystem). What is known of the subject at the present time is insufficient in that it suffers from mathematical restrictions which are difficult to justify in practice, is limited in the types of noises that can be accommodated, which do not cover the noises one meets in nature, and is based on algorithms whose behavior is not sufficiently understood. The broad aim of this thesis is to solve some of the problems that are open in that area of research. As detection of a non-Gaussian stochastic signal in additive and dependent Gaussian noise can be viewed as the canonical detection problem for active sonar in a reverberation-limited environment, and that this detection problem, except for a multiplicity restriction, is, mathematically, the problem nearest to a satisfactory solution, the first part of the thesis deals with the definition and the properties of a form of the Itô stochastic integral that must be tailored to remove the multiplicity restriction mentioned above. On the way some interesting connections with other forms of the stochastic integral are investigated. The second part of the thesis is devoted to the derivation of the likelihood ratio which acts as an universal detector, still within the framework of a stochastic signal in dependent Gaussian noise. The solution of the Gaussain noise problem is based on a representation of the noise and signal-plus-noise processes as superpositions of causal filters acting on noise and signal-plus-noise processes that are semimartingales, for the treatment of which stochastic calculus is the most efficient tool. The decomposition used is the Cramér-Hida decomposition which is particularly suited to the handling of Gaussian processes, though it is much more broadly valid. The last part of the thesis is a study of the possibility to extend the method that works for Gaussian noise to situations for which the noise is no longer Gaussian. A likelihood formula is obtained for noises that are non anti- cipative transformations of the sum of a Wiener process and an independent Poisson martingale.