Define the premium for a risk $X\geq0$ as:
$$ π(X)=∫^∞_0(1−F_X(t))^{1/ρ}dt $$
With $\rho\geq1$.
Let $X,Y\geq0$ two independent random variables. Prove or find a counterexample for:
$$ \pi(X+Y)\leq\pi(X)+\pi(Y) $$
I've tried a lot of counterexamples, but any of them work. It looks that it actually works, but I don't know how to prove it. Please help!
UPDATE:
Stochastic Processes for Insurance and Finance (page 89)
Tomasz Rolski, Hanspeter Schmidli, V.
It is left to the reader as an exercise. So it is actually true. How to prove it?