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.
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.
@article{AFST_2006_6_15_1_125_0, author = {Wschebor, Mario}, title = {Smoothing and occupation measures of stochastic processes}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {125--156}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 15}, number = {1}, year = {2006}, doi = {10.5802/afst.1116}, zbl = {1121.62072}, mrnumber = {2225750}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1116/} }
TY - JOUR AU - Wschebor, Mario TI - Smoothing and occupation measures of stochastic processes JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2006 SP - 125 EP - 156 VL - 15 IS - 1 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1116/ DO - 10.5802/afst.1116 LA - en ID - AFST_2006_6_15_1_125_0 ER -
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Wschebor, Mario. Smoothing and occupation measures of stochastic processes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 1, pp. 125-156. doi : 10.5802/afst.1116. http://archive.numdam.org/articles/10.5802/afst.1116/
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