Bellman equation for terminal-wealth utility
= Bellman equation for terminal-wealth utility
For terminal utility $U$ and independent return innovations,
$$
V(N,x)=U(x),\qquad
V(n,x)=\sup_{\theta\in\mathbb R^d}
\mathbb E\left[V\left(n+1,(1+r)x+\theta^T\xi_{n+1}\right)\right].
$$