represent every local martingale as a combination of two components, namely as an stochastic integral with respect to the continuous part of the semimartingale and an integral with respect to its compensated jump measure. This is of course the case for local martingales, and with more reason this observation about the dimensionality holds for the martingales associated with the corresponding densities processes. In this section we review those concepts of stochastic calculus that are relevant to understand this representation properties, and prove some kind of continuity property for the quadratic variation of a sequence of uniformly integrable martingales converging in  $L^{1}$. This result is one of the contributions of this paper.

Fundamentals of Lévy and semimartingales processes [Subsect:\_Fundamentals\_Levy\_and\_Semimartingales]{}
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Let $\left( \Omega ,\mathcal{F},\mathbb{P}\right) $ be a probability space. We say that $L:=\left\{ L_{t}\right\} _{t\in \mathbb{R}_{+}}$ is a Lévy process for this probability space if it is an adapted càdlàg process with independent stationary increments starting at zero. The filtration considered is $\mathbb{F}:=\left\{ \mathcal{F}_{t}^{\mathbb{P}}\left( L\right) \right\} _{t\in \mathbb{R}_{+}}$, the completion of its natural filtration, i.e. $\mathcal{F}_{t}^{\mathbb{P}}\left( L\right)
:=\sigma \left\{ L_{s}:s\leq t\right\} \vee \mathcal{N}$ where $\mathcal{N}$ is the $\sigma $-algebra generated by all $\mathbb{P}$-null sets. The jump measure of $L$ is denoted 