Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 2 / c

Use $w$ for current wealth and $\pi =w-B$ for the dollar position in the risky asset. Its controlled dynamics have drift $rw+(\mu -r)\pi$ and diffusion coefficient $\sigma \pi$. Over a short time step $h$, the dynamic programming principle givesApply Itô formula to the second term, divide by $h$, and let $h\downarrow 0$. The Hamilton-Jacobi-Bellman equation isFor the increasing concave value function, $J_w>0$ and $J_{ww}<0$, so the expression is a strictly concave quadratic in the risky position. Its derivative vanishes atSubstitution of this maximizer yieldsThere is a running utility reward but no consumption withdrawal in the wealth dynamics. The required concavity is also verified explicitly by the positive coefficient in part (d), so this stationary point really is the maximum.