= Solution
Set $V(n,x)=A_n+B_nx+C_nx^2$. The terminal values are $(A_N,B_N,C_N)=(0,0,1)$. Since the noise is centred, backward induction keeps $B_n=0$. Completing the square in the Bellman equation gives
$$
u^*=-\frac{C_n}{1+C_n}x,qquad
C_{n-1}=\frac{C_n}{1+C_n},qquad
A_{n-1}=A_n+C_n\sigma^2.
$$
Hence
$$
\boxed{C_n=\frac1{N-n+1},\quad B_n=0,\quad
A_n=\sigma^2\sum_{j=n+1}^{N}\frac1{N-j+1}}
$$
(with the empty sum equal to zero). This proves the asserted quadratic form of the <value function>.
Back to article page
