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In papers by Yor, a remarkable class $(\Sigma)$ of submartingales isintroduced, which, up to technicalities, are submartingales $(X_{t})_{t\geq0}$whose increasing process is carried by the times $t$ such that $X_{t}=0$. Thesesubmartingales have several applications in stochastic analysis: for example,the resolution of Skorokhod embedding problem, the study of Brownian localtimes and the study of zeros of continuous martingales. The submartingales ofclass $(\Sigma)$ have been extensively studied in a series of articles byNikeghbali (part of them in collaboration with Najnudel, some others withCheridito and Platen). On the other hand, stochastic calculus has been extendedto signed measures by Ruiz de Chavez \cite{chav} and Beghdadi-Sakrani\cite{sak}. In \cite{f}, the authors of the present paper have extended thenotion of submartingales of class $(\Sigma)$ to the setting of Ruiz de Chavez\cite{chav} and Beghdadi-Sakrani \cite{sak}, giving two different classes ofstochastic processes named classes $\sum(H)$ and $\sum_{s}(H)$ where from toolsof the theory of stochastic calculus for signed measures, the authors providegeneral frameworks and methods for dealing with processes of these classes. Inthis work, we first give some formulas of multiplicative decomposition forprocesses of these classes. Afterward, we shall establish some representationresults allowing to recover any process of one of these classes from its finalvalue and the last time it visited the origin.

On the study of processes of classes $\sum(H)$ and $\sum_{s}(H)$