I'm trying to get $E(\max \{ a-X, b-X-Y, 0 \})$,
where $X$ ~ $N(0,\sigma^2)$, and $Y$ ~ $N(\mu, \gamma^2)$, and $X,Y$ are independent.
I've been trying to figure this out by doing,
$E(\max \{ a-X, b-X-Y, 0 \}) = E(\max \{ a-X, 0 \}|Y \geq b-X)P(Y \geq b-X) + E(\max \{ a-X, b-X-Y \}|Y < b-X)P(Y < b-X)$,
but the second part of the RHS is hard to solve. Are there any tricks that I can use to solve this problem? Thank you.