Natural decomposition of processes and weak Dirichlet processes

Jean Mémin (1) , François Coquet (2) , Adam Jakubowski (3) , Leszek Slominski (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMAH - Laboratoire de Mathématiques Appliquées du Havre
3 FMCS - Faculty of Mathematics and Computer Science [Toruń]

A class of stochastic processes, called "weak Dirichlet processes", is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C^1$-transformations and under absolutely continuous change of measure. If a weak Dirichlet process has finite energy, as defined by Graversen and Rao, its Doob-Meyer type decomposition is unique. The developed methods have been applied to a study of generalized martingale convolutions.

Mots clés: weak Dirichlet processes; Dirichlet processes; semimartingale; quadratic variation; Ito's formula; generalized martingale convolution

Domaines: Probabilités [math.PR]

Jean Mémin, François Coquet, Adam Jakubowski, Leszek Slominski. Natural decomposition of processes and weak Dirichlet processes. 2004. ⟨hal-00001360v1⟩