Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 30K / b / Solution

Set $V(n,x)=A_n+B_{n}x+C_n{x}^2$. 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 givesHence(with the empty sum equal to zero). This proves the asserted quadratic form of the value function.