the variation of $A$ over the finite interval $\left[ 0,t\right] $. The subset $\mathcal{A}^{+}=\mathcal{A\cap V}^{+}$ represents those processes which are also increasing i.e. with non-negative right-continuous increasing trajectories. Furthermore, $\mathcal{A}_{loc}$ (resp. $\mathcal{A}_{loc}^{+}$) is the collection of adapted processes with locally integrable variation (resp. adapted locally integrable increasing processes). For a càdlàg process $X$ we denote by $X_{-}:=\left( X_{t-}\right) $ the left hand limit process, where $X_{0-}:=X_{0}$ by convention, and by $\bigtriangleup X=\left( \bigtriangleup X_{t}\right) $ the jump process $\bigtriangleup X_{t}:=X_{t}-X_{t-}$.

Given an adapted càdlàg semimartingale $U$, the jump measure and its dual predictable projection (or compensator) are denoted by $\mu _{U}\left( \left[ 0,t\right] \times A\right) :=\sum_{s\leq t}\mathbf{1}_{A}\left(
\triangle U_{s}\right) $ and $\mu _{U}^{\mathcal{P}}$, respectively. Further, we denote by $\mathcal{P}\subset \mathcal{F}\otimes \mathcal{B}\left( \mathbb{R}_{+}\right) $ the predictable $\sigma $-algebra and by $\widetilde{\mathcal{P}}:=\mathcal{P}\otimes \mathcal{B}\left( \mathbb{R}_{0}\right) .$ With some abuse of notation, we write $\theta _{1}\in
\widetilde{\mathcal{P}}$ when the function $\theta _{1}:$ $\Omega \times
\mathbb{R}_{+}\times \mathbb{R}_{0}\rightarrow \mathbb{R}$ is $\widetilde{\mathcal{P}}$-measurable and $\theta \in \mathcal{P}$ for predictable processes.

Let $$\begin{array}{clc}
\mathcal{L}\left( U^{c}\right) := & \left\{ \theta \in \mathcal{P}:\exists
\left\{ \tau _{n}\right\} _{n\in \mathbb{N}}\text{ sequence of stopping
times with }\tau _{n}\uparrow \infty \right. &  \\
& \left. \text{and }\mathbb{E}\left[ \int\limits_{0}^{\tau _{n}}\theta ^{2}d\left[ U^{c}\right] \right] <\infty \ \forall 