I am working through the textbook "Modern Actuarial Risk Theory" and came across the part where stop loss orders are introduced. It is defined as $X<_{SL}Y \iff E[(X-d)_{+}] <= E[(Y-d)_{+}], d\geq0$.
Now let X,Z be two risks with $X<_{SL}Z$, define $Y=max(X,b), b\geq0$. Find the $b\geq0$ such that $E[Y]=E[Z]$.
I guess this b is somehow related to $E[Z]$, but can't think of something. So far I've come up with $P(Y\leq y)=0$ for $y<b$ and $P(Y \leq y)=P(X\leq y)$ elsewhere, but even after using this to calculate $E[Y]$ I cannot find a suitable b.
Does anyone have any hints for me?