càdlàg uniformly integrable martingale respectively. The following lemma is interesting by itself to understand the continuity properties of the quadratic variation for a given convergent sequence of uniformly integrable martingale . It will play a central role in the proof of the lower semicontinuity of the penalization function introduced in section \[Sect Penalty Function for densities\]. Observe that the assertion of this lemma is valid in a general filtered probability space and not only for the completed natural filtration of the Lévy process introduced above.

\[E\[|Mn-M|\]-&gt;0=&gt;\[Mn-M\](oo)-&gt;0\_in\_P\]For $\left\{ M^{\left( n\right)
}\right\} _{n\in \mathbb{N}}\subset \mathcal{M}_{\infty }$ and $M\in
\mathcal{M}_{\infty }$ the following implication holds $$M_{\infty }^{\left( n\right) }\overset{L^{1}}{\underset{n\rightarrow \infty }{\longrightarrow }}M_{\infty }\Longrightarrow \left[ M^{\left( n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\longrightarrow }0.$$Moreover,$$M_{\infty }^{\left( n\right) }\overset{L^{1}}{\underset{n\rightarrow \infty }{\longrightarrow }}M_{\infty }\Longrightarrow \left[ M^{\left( n\right) }-M\right] _{t}\overset{\mathbb{P}}{\underset{n\rightarrow \infty }{\longrightarrow }}0\;\; \forall t.$$

*Proof.* From the $L^{1}$ convergence of $M_{\infty
}^{\left( n\right) }$ to $M_{\infty }$, we have that $\{M_{\infty }^{\left(
n\right) }\}_{n\in \mathbb{N}}\cup \left\{ M_{\infty }\right\} $ is uniformly integrable, which is equivalent to the existence of a convex and increasing function $G:[0,+\infty )\rightarrow \lbrack 0,+\infty )$ such that $$\left( i\right) \quad \lim_{x\rightarrow \infty }\frac{G\left( x\right) }{x}=\infty ,$$and $$\left( ii\right) \quad \sup_{n\in \mathbb{N}}\mathbb{E}\left[ G\left(
\left\vert M_{\infty }^{\left( n\right) 