by $\mu :\Omega \times \left( \mathcal{B}\left(
\mathbb{R}_{+}\right) \otimes \mathcal{B}\left( \mathbb{R}_{0}\right)
\right) \rightarrow \mathbb{N}$ where $\mathbb{R}_{0}:=\mathbb{R}\setminus
\left\{ 0\right\} $. The dual predictable projection of this measure, also known as its Lévy system, satisfies the relation $\mu ^{\mathcal{P}}\left( dt,dx\right) =dt\times \nu \left( dx\right) $, where $\nu \left(
\cdot \right) :=\mathbb{E}\left[ \mu \left( \left[ 0,1\right] \times \cdot
\right) \right] $ is the intensity or Lévy measure of $L.$

The Lévy-Itô decomposition of $L$ is given by $$L_{t}=bt+W_{t}+\int\limits_{\left[ 0,t\right] \times \left\{ 0<\left\vert
x\right\vert \leq 1\right\} }xd\left\{ \mu -\mu ^{\mathcal{P}}\right\}
+\int\limits_{\left[ 0,t\right] \times \left\{ \left\vert x\right\vert
>1\right\} }x\mu \left( ds,dx\right) .  \label{Levy-Ito_decomposition}$$It implies that $L^{c}=W$ is the Wiener process, and hence $\left[ L^{c}\right] _{t}=t$, where $\left( \cdot \right) ^{c}$ and $\left[ \,\cdot \,\right] $ denote the continuous martingale part and the process of quadratic variation of any semimartingale, respectively. For the predictable quadratic variation we use the notation $\left\langle \,\cdot \,\right\rangle $.

Denote by $\mathcal{V}$ the set of càdlàg, adapted processes with finite variation, and let $\mathcal{V}^{+}\subset \mathcal{V}$ be the subset of non-decreasing processes in $\mathcal{V}$ starting at zero. Let $\mathcal{A}\subset \mathcal{V}$ be the class of processes with integrable variation, i.e. $A\in \mathcal{A}$ if and only if $\bigvee_{0}^{\infty }A\in
L^{1}\left( \mathbb{P}\right) $, where $\bigvee_{0}^{t}A$ denotes 