maybe someone can help me to prove the coherence properties for a risk measure.
It's about the coherence of the following risk measure (Entropic Value-at-Risk): $EVaR_\alpha(X):=\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{M_X(z)}{1-\alpha} \right) \}$ where $M_X(z)$ is the moment generating function.
The coherence properties which I want to prove are the following:
Monotonicity: For $X_i$, $X_j\in V$ with $X_i \leq X_j$ $$ \rho(X_i)\leq \rho(X_j) $$ Translation invariance: For all $X_i\in V$ and a constant $c\in\mathbb{R}$ $$ \rho(X_i+c)=\rho(X_i)+c $$ Positive homogeneity: For all $X_i\in V$ and a constant $\lambda \geq 0$ $$ \rho(\lambda \cdot X_i)=\lambda \cdot\rho(X_i) $$ Sub-additivity: For $X_i$, $X_j\in V$ $$ \rho(X_i+X_j)\leq\rho(X_i)+\rho(X_j) $$
For example if I want to prove the Translation invariance: $$\rho(X+c)=\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{M_{X+c}(z)}{1-\alpha} \right) \} =\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{z(X+c)})}{1-\alpha} \right) \} $$ Can I write the expected value as a product because of the independency? $$ =\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{zX})E(e^{zc})}{1-\alpha} \right) \}=\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{zX})e^{zc}}{1-\alpha} \right) \} =\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{zX})}{1-\alpha} \right)+\frac{1}{z}\ln(e^{zc}) \}\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{zX})}{1-\alpha} \right)\}+c=\rho(X)+c $$ Does it work like this? If yes: how can I prove the Positive homogeneity? $$ \rho(\lambda \cdot X)=\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{M_{X\lambda}(z)}{1-\alpha} \right) \} =\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{zX\lambda})}{1-\alpha} \right) \} $$ I don't know how to go on at this point? I would like to write something like this: $$ =\underset{z>0}{\inf}\{\frac{1}{z}\lambda\ln\left( \frac{E(e^{Xz})}{1-\alpha} \right) \}=\lambda\underset{z>0}{\inf}\{\frac{1}{z}\ln\left( \frac{E(e^{Xz})}{1-\alpha} \right) \}=\lambda \rho( X) $$ But I can't use the logarithmic laws/identities, because the expected value is a sum?
I hope someone can help me. Any help would be greatly appreciated! :)
Best regards
