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 be a risk measure so that
- is a nonatomic probability space
- is law-invariant, i.e., if are two random variables with the same probability distribution, then
- the risk measure is coherent (i.e., it is convex, monotonous, translation invariant and positive homogeneous)
- it is also lower semicontinuous
Take a nonnegative random variable (a.s.). Then if and only if (a.s.).
The proof is quite straightforward and hinges on the fact that under the prescribed assumptions
for all . Assume that . Then, which implies that , thus is almost everywhere zero.
Conversely, if , we may take the random variable which is everywhere zero ( for all ). Then and have the same distribution, so .