New classes of processes in stochastic calculus for signed measures

TL;DR

This paper introduces new classes of processes within stochastic calculus for signed measures, extending the existing class (∑) by characterizing processes with specific local martingale and non-decreasing properties relative to a signed measure framework.

Contribution

It defines two novel classes of processes in the context of signed measures, generalizing the class (∑) and providing a framework for their analysis.

Findings

Extended the class (∑) to new process classes

Characterized processes with specific zero-set properties

Provided theoretical foundations for stochastic calculus with signed measures

Abstract

Let us consider a signed measure $\Qv$ and a probability measure $\Pv$ such that $\Qv<<\Pv$. Let $D$ be the density of $\Qv$ with respect to $\Pv$. $H$ represents the set of zeros of $D$, $\bar{g}=0\vee\sup{H}$. 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 $ND$ is a cadlag local martingale and $A$ is a continuous and non-decreasing process such that $(dA_{t})$ is carried by $H\cup\{t: X_{t}=0\}$. The second one is the case where $N$ and $A$ are null on $H$ and $A_{.+\bar{g}}$ is a non-decreasing, continuous process such that $(dA_{t+\bar{g}})$ is carried by $\{t: X_{t+\bar{g}}=0\}$. We shall show that these classes are extensions of the class $(\sum)$ defined by A.Nikeghbali \cite{nik} in the framework of stochastic calculus for signed measures.