On the Study of Processes of $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) Classes

In papers by Yor, a remarkable class $$({\varSigma })$$ ( Σ ) of submartingales is introduced, which, up to technicalities, are submartingales $$(X_{t})_{t\ge 0}$$ ( X t ) t ≥ 0 whose increasing process is carried by the times t such that $$X_{t}=0$$ X t = 0 . These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class $$({\varSigma })$$ ( Σ ) have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le théorème de Paul Lévy pour des mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class $$({\varSigma })$$ ( Σ ) to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin.