Smoothing and occupation measures of stochastic processes

Wschebor, Mario

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 125-156.

Cet article est une révision d’un certain nombre de problèmes statistiques concernant les processus aléatoires à un paramètre continu. En général, on suppose que l’observable est une régularisation de la trajectoire du processus, obtenue par convolution avec un noyau détérministe. La plupart des résultats ici exposés est connue et presentée sans démonstration. Les énoncés des théorèmes contiennent des approximations de la mesure d’occupation, au premier et deuxième ordre, basées sur des fonctionnelles définies sur les régularisées des trajectoires. On considère diverses classes de processus, à savoir, le processus de Wiener, les processus gaussiens, les semi-martingales continues et les processus de Lévy. Nous avons inclus les détails de certaines applications statistiques.

This is a review paper about some problems of statistical inference for one-parameter stochastic processes, mainly based upon the observation of a convolution of the path with a non-random kernel. Most of the results are known and presented without proofs. The tools are first and second order approximation theorems of the occupation measure of the path, by means of functionals defined on the smoothed paths. Various classes of stochastic processes are considered starting with the Wiener process, Gaussian processes, continuous semi-martingales and Lévy processes. Some statistical applications are also included in the text.

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