Let $U$ be concave and strictly increasing, and let $\mathcal{A}$ be a vector space of random variables. For every random variable $Z$ let

$$F(Z)=\sup _{X \in \mathcal{A}} \mathbb{E}[U(X+Z)]$$

and suppose there exists a random variable $X_{Z} \in \mathcal{A}$ such that

$$F(Z)=\mathbb{E}\left[U\left(X_{Z}+Z\right)\right]$$

For a random variable $Y$, let $\pi(Y)$ be such that $F(Y-\pi(Y))=F(0)$.

(a) Show that for every constant $a$ we have $\pi(Y+a)=\pi(Y)+a$, and that if $\mathbb{P}\left(Y_{1} \leqslant Y_{2}\right)=1$, then $\pi\left(Y_{1}\right) \leqslant \pi\left(Y_{2}\right)$. Hence show that if $\mathbb{P}(a \leqslant Y \leqslant b)=1$ for constants $a \leqslant b$, then $a \leqslant \pi(Y) \leqslant b .$

(b) Show that $Y \mapsto \pi(Y)$ is concave, and hence show $t \mapsto \pi(t Y) / t$ is decreasing for $t>0$.

(c) Assuming $U$ is continuously differentiable, show that $\pi(t Y) / t$ converges as $t \rightarrow 0$, and that there exists a random variable $X_{0}$ such that

$$\lim _{t \rightarrow 0} \frac{\pi(t Y)}{t}=\frac{\mathbb{E}\left[U^{\prime}\left(X_{0}\right) Y\right]}{\mathbb{E}\left[U^{\prime}\left(X_{0}\right)\right]}$$