# Coherent risk measure and its dual representation

We will discuss the definition of coherent risk measures and its dual representation in this below. See its summary in [Rud07].

Let ${(\Omega, \mathcal{F}, P)}$ be a complete probability space. Let ${\mathcal{P}}$ be a probability measure space, and ${\mathcal{P}_\infty = \{Q \in \mathcal{P}: dQ/ dP \in L^\infty \}}$. A function ${\rho: L^1 \mapsto \mathbb{R} \cup\{+\infty\}}$ is called coherent risk measure, if it satisfies the following conditions:

1. Normalization: ${\rho(0) = 0}$;
2. Monotonicity: ${\rho(X_1) \le \rho(X_2)}$ whenever ${X_1 \ge X_2}$;
3. Translation invariance: ${\rho(X +c) = \rho(X) - c}$ for any constant ${c}$;
4. Subadditivity: ${\rho(X_1 + X_2) \le \rho(X_1) + \rho(X_2)}$;
5. Positive homogeneity: ${\rho(\alpha X) = \alpha \rho(X)}$ for any ${\alpha \ge 0}$;

Proposition 1 ${\rho}$ is a weak* lower semicontinuous coherent risk measure if and only if there exists a non-empty subset of a probability measures ${\mathcal{\tilde Q}}$ with ${\Big\{\frac{d {Q}}{d {P}}: Q\in \mathcal{\tilde Q} \Big\}}$ convex and weak* closed in ${L^\infty}$, such that

$\displaystyle \rho(X) = \sup_{Q\in \mathcal{\tilde Q}} \mathbb{E}^Q[-X].$