# 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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