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].

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