Zero risk

What does it mean for a random variable to exhibit zero risk? It of course depends on the risk measure we are using to quantify it.

Let \rho:\mathcal{L}_p(\Omega, \mathscr{F}, \mathrm{P}) be a risk measure so that

  1. (\Omega, \mathscr{F}, \mathrm{P}) is a nonatomic probability space
  2. \rho is law-invariant, i.e., if X, Y are two random variables with the same probability distribution, then \rho(X) = \rho(Y)
  3. the risk measure is coherent (i.e., it is convex, monotonous, translation invariant and positive homogeneous)
  4. it is also lower semicontinuous

Take a nonnegative random variable Z\geq 0 (a.s.). Then \rho(Z)=0 if and only if Z=0 (a.s.).

The proof is quite straightforward and hinges on the fact that under the prescribed assumptions

\mathbb{E}[X] \leq \rho[X]

for all X. Assume that \rho[Z]=0. Then, 0 \leq \mathbb{E}[Z] \leq \rho[Z] = 0 which implies that \mathbb{E}[Z]=0, thus Z is almost everywhere zero.

Conversely, if Z=0, we may take the random variable Z' which is everywhere zero (Z(\omega)=0 for all \omega\in\Omega). Then Z and Z' have the same distribution, so \rho[Z]=\rho[Z']=0.

 

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