New classes of processes in stochastic calculus for 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 $(d{A}_t)$ 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 $(d{A}_{t+\bar{g}})$ 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.

23 pages. arXiv admin note: text overlap with arXiv:math/0505515