results from the theory of static risk measures, and then a characterization for minimal penalties is presented.

Preliminaries from static measures of risk [Subsect:\_Preliminaries\_SCRM]{}
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Let $X:\Omega \rightarrow \mathbb{R}$ be a mapping from a set $\Omega $ of possible market scenarios, representing the discounted net worth of the position. Uncertainty is represented by the measurable space $(\Omega,
\mathcal{F})$, and we denote by $\mathcal{X}$ the linear space of bounded financial positions, including constant functions.

1.  The function $\rho :\mathcal{X}\rightarrow \mathbb{R}$, quantifying the risk of $X$, is a *monetary risk measure* if it satisfies the following properties: $$\begin{array}{rl}
    \text{Monotonicity:} & \text{If }X\leq Y\text{ then }\rho \left( X\right)
    \geq \rho \left( Y\right) \ \forall X,Y\in \mathcal{X}.\end{array}
    \label{Monotonicity}$$$\smallskip \ $$$\begin{array}{rl}
    \text{Translation Invariance:} & \rho \left( X+a\right) =\rho \left(
    X\right) -a\ \forall a\in \mathbb{R}\ \forall X\in \mathcal{X}.\end{array}
    \label{Translation Invariance}$$

2.  When this function satisfies also the convexity property $$\begin{array}{rl}
    & \rho \left( \lambda X+\left( 1-\lambda \right) Y\right) \leq \lambda \rho
    \left( X\right) +\left( 1-\lambda \right) \rho \left( Y\right) \ \forall
    \lambda \in \left[ 0,1\right] \ \forall X,Y\in \mathcal{X},\end{array}
    \label{Convexity}$$it is said that $\rho $ is a convex risk measure.

3.  The function $\rho $ is called normalized if $\rho \left(
    0\right) =0$, and sensitive, with 