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

TL;DR

This paper introduces a new class of stochastic processes, $\\Sigma^{r}(H)$, unifying two known classes, and explores their properties, representations, and applications, including solutions for skew Brownian motion equations.

Contribution

The paper defines the class $\\Sigma^{r}(H)$, unifying classes $(\Sigma)$ and $\mathcal{M}(H)$, and provides characterization, representation, and application results.

Findings

Unified framework for classes $(\Sigma)$ and $\mathcal{M}(H)$

Representation formulas from final values and honest times

Application to skew Brownian motion solutions

Abstract

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 $\Sigma^{r}(H)$.