A new class of stochastic processes with great potential for interesting applications

This paper contributes to the study of a new and remarkable family of stochastic processes that we will term class $\Sigma^{r}(H)$. This class is potentially interesting because it unifies the study of two known classes: the class $(\Sigma)$ and the class $\mathcal{M}(H)$. In other words, we consider the stochastic processes $X$ which decompose as $X=m+v+A$, where $m$ is a local martingale, $v$ and $A$ are finite variation processes such that $dA$ is carried by $\{t\geq0:X_{t}=0\}$ and the support of $dv$ is $H$, the set of zeros of some continuous martingale $D$. First, we introduce a general framework. Thus, we provide some examples of elements of the new class and present some properties. Second, we provide a series of characterization results. Afterwards, we derive some representation results which permit to recover a process of the class $\Sigma^{r}(H)$ from its final value and of the honest times $g=\sup\{t\geq0:X_{t}=0\}$ and $\gamma=\sup{H}$. In final, we investigate an interesting application with processes presently studied. More precisely, we construct solutions for skew Brownian motion equations using stochastic processes of the class $\Sigma^{r}(H)$.Comment: 23 page

Approche probabiliste des particules collantes et système de gaz sans pression

Président du jury: Davidov, YouriAt each time $t$, our construction of sticky particle dynamics, with initial mass repartition function $F_0$ and velocities $u_0$, is given by the convex hull $H(\cdot,t)$ of the function $m\in (0, 1)\mapsto \int_a^m\big( F_0^(-1)(z) + tu_0\big(F_0^(-1)(z)\big)\big)dz$. Here, $F_0^(-1)$ is one of two inverse functions of $F_0$. We show that the processes $X_t^-(m)= \partial_m^-H(m,t),\; X_t^+(m) = \partial_m^+H(m,t)$, defined on the probability space $([0, 1], (\cal B), \lambda)$, are not distinguishable, and they model the particle trajectories. The process $X_t=:X_t^-=X_t^+$ is a solution of the equation (SDE):\;\frac(dX_t)(dt) =\E[ u_0(X_0)/X_t], such that $P(X_0 \leq x) = F_0(x)\,\,\forall x$. The inverse $M_t:= M(\cdot,t)$ of the map $m\mapsto \partial_mH(m,t)$ is the mass repartition function at time $t$. It is also the repartition function of the random variable $X_t$. We show the existence of a forward flow map $\big(\phi(x,t,M_s, u_s),\,s 0\big)$ is a weak solution of the pressure-less gas system of equations with initial datum $\frac(dF_0(x))(dx), u_0$. This thesis also presents other solutions of the above stochastic differential equation $(SDE)$.A chaque instant $t$, nous construisons la dynamique des particules collantes dont la masse est distribuée initialement suivant une fonction de répartition $F_0$, avec une vitesse $u_0$, à partir de l'enveloppe convexe $H(\cdot,t)$ de la fonction $m\in (0,1)\mapsto \int_a^m\big( F_0^(-1)(z) + tu_0\big(F_0^(-1)(z)\big)\big)dz$. Ici, $F_0^(-1)$ est l'une des deux fonctions inverses de $F_0$. Nous montrons que les deux processus stochastiques $X_t^-(m)= \partial_m^-H(m,t),\; X_t^+(m) = \partial_m^+H(m,t)$, définis sur l'espace probabilisé $([0, 1], (\cal B), \lambda)$, sont indistinguables et ils modélisent les trajectoires des particules. Le processus $X_t:= X_t^- = X_t^+$ est une solution de l'équation (EDS): \; \frac(dX_t)(dt) =\E[ u_0(X_0)/X_t], telle que $P(X_0 \leq x) = F_0(x)\,\,\forall x$. L'inverse $M_t:= M(\cdot,t)$ de la fonction $m\mapsto \partial_mH(m,t)$ est la fonction de répartition de la masse à l'instant $t$. Elle est aussi la fonction de répartition de la variable aléatoire $X_t$. On montre l'existence d'un flot $(\phi(x,t,M_s, u_s))_( s 0)$ est une solution faible du système de gaz sans pression de données initiales $\frac(dF_0(x))(dx), u_0$. Cette thèse contient aussi d'autres solutions de l'équation différentielle stochastique $(EDS)$ ci-dessus