This paper consists of two independent parts. In the first one, we contribute to the study of the class (Σ). For instance, we provide a new way to characterize stochastic processes of this class. We also present some new properties and solve the Bachelier equation. In the second part, we study the class of stochastic processes Σ(H). This class was introduced in [7] where from tools of the theory of martingales with respect to a signed measure of [21], the authors provide a general framework and methods for dealing with processes of this class. In this work, after developing some new properties, we embed a non-atomic measure ν in X, a process of the class Σ(H). More precisely, we find a stopping time T < ∞ such that the law of X T is ν. with respect to a signed measure; Hardy-Littlewood function MSC:60G07; 60G20; 60G46; 60G48 1 F. EYI-OBIANG et al dV t carried by the set {t ≥ 0 : X t = 0}. It plays an important role in martingale theory. For instance: the family of Azéma-Yor martingales, the resolution of Skorokhod's embedding problem, the study of Brownian local times. It also plays a key role in the study of zeros of continuous martingales. A large class containing all the previously mentioned processes is the class (Σ) whose definition is given in section 2. This class was introduced by Yor in [24] but some of its main properties were further studied in [6,12,13,14,15,16,17].Recently, the study of stochastic processes of the form of Equation (1) was generalized in the field of stochastic calculus for signed measures. That is, the stochastic calculus when the measure space is governed, not by a probability measure, but by a general measure that can take positive and negative values (signed measure). In particular, a new class of processes satisfying Equation (1) was introduced in [7] where the authors provide a general framework and methods based on the tools of the martingale theory for signed measures developed by Ruiz de Chavez in [21]. This class is called, class Σ(H) and we shall define it in Section 3.The aim of this paper is to bring some contributions in frameworks of two above mentioned classes of stochastic processes. More precisely, the Section 2 is dedicated to the study of the class (Σ) and Section 3 is reserved to the study of the class Σ(H). In particular, in Section 2, we shall provide a new characterization of processes of class (Σ). We give also a new solution of Bachelier equation. In Section 3, we prove the following estimates for processes of the class Σ(H), generalizing a well known result for the pair (X t , A t ) and for random variable A ∞ where X is a positive submartingale of the class (Σ) and A its non-decreasing process:

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Let us consider a signed measure Q and a probability measure P such that Q << P. Let D be the density of Q with respect to P. H represents the set of zeros of D, g = 0 ∨ sup H. In this paper, we shall consider two classes of nonnegative processes of the form X t = N t + A t . The first one is the class of semimartingales where N D is a cadlag local martingale and A is a continuous and non-decreasing process such that (dA t ) is carried by H ∪ {t : X t = 0}. The second one is the case where N and A are null on H and A .+g is a non-decreasing, continuous process such that (dA t+g ) is carried by {t : X t+g = 0}. We shall show that these classes are extensions of the class ( ) defined by A.Nikeghbali [6] in the framework of stochastic calculus for signed measures.

Under general assumptions on the initial data, we show that the entropy solution (x, t) ↦ u(x, t) of the one-dimensional inviscid Burgers’ equation is the velocity function of a sticky particles model whose initial mass distribution is Lebesgue measure. Precisely, the particles trajectories (x, t) ↦ X0, t(x) are given by a forward flow: \documentclass[12pt]{minimal}\begin{document}$\forall \,(x,s,t)\in \mathbb {R}\times \mathbb {R}_+\times \mathbb {R}_+, X_{0,s+t}(x)=X_{s,t}\big (X_{0,s}(x)\big )\;\mbox{ and}\; \frac{\partial }{\partial t} X_{s,t}\break=u(X_{s,t},s+t)=\mathrm{E}[u(\cdot ,s)|X_{s,t}];\,\, u(x,t)=\mathrm{E}[u(\cdot ,0)|X_{0,t}=x].$\end{document}∀(x,s,t)∈R×R+×R+,X0,s+t(x)=Xs,tX0,s(x)and∂∂tXs,t=u(Xs,t,s+t)=E[u(·,s)|Xs,t];u(x,t)=E[u(·,0)|X0,t=x].

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