_{\rho
    }^{\ast }\left( \mathbb{Q}\right) =+\infty =\psi \left( \mathbb{Q}\right).$$ Therefore, under the conditions of Lemma \[static minimal penalty funct. in Q(&lt;&lt;) &lt;=&gt;\] $\left( b\right) $ the penalty function $\psi $ might differ from $\psi _{\rho
    }^{\ast }$ on $\mathcal{Q}_{cont}^{\ll }\setminus \mathcal{Q}_{\ll }.$ For instance, the penalty function defined as $\psi \left( \mathbb{Q}\right) :=\infty \times \mathbf{1}_{\mathcal{Q}_{cont}\setminus  \mathcal{Q}_{\ll }(\mathbb{P})}\left( \mathbb{Q}\right) $ leads to the worst case risk measure $\rho (X):=\sup\nolimits_{\mathbb{Q}\in \mathcal{Q}_{\ll }(\mathbb{P})}\mathbb{E}_{\mathbb{Q}}\left[ -X\right] $, which has as minimal penalty the function $$\psi _{\rho }^{\ast
    }\left( \mathbb{Q}\right) =\infty \times \mathbf{1}_{\mathcal{Q}_{cont}\setminus \mathcal{Q}_{cont}^{\ll }}\left( \mathbb{Q}\right).$$

2.  Note that the total variation distance $d_{TV}\left( \mathbb{Q}^{1},\mathbb{Q}^{2}\right) :=\sup_{A\in \mathcal{F}}\left\vert \mathbb{Q}^{1}\left[ A\right] -\mathbb{Q}^{2}\left[ A\right] \right\vert $, with $\mathbb{Q}^{1},\;\mathbb{Q}^{2}\in \mathcal{Q}_{\ll }$, fulfills that $d_{TV}\left( \mathbb{Q}^{1},\mathbb{Q}^{2}\right) \leq \left\Vert d\mathbb{Q}^{1}/d\mathbb{P}-\mathbb{Q}^{2}/d\mathbb{P}\right\Vert _{L^{1}}$. Therefore, the minimal penalty function is lower semicontinuous in the total variation topology; see Remark 4.16 (b) p. 163 in [@FoellSch; @2004].

Preliminaries from stochastic calculus\[Sect. Preliminaries\]
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Within a probability space which supports a semimartingale with the weak predictable representation property, there is a representation of the density processes of the absolutely continuous probability measures by means of two coefficients. Roughly speaking, this means that the dimension of the linear space of local martingales is two. Throughout these coefficients we can 