Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 39 / 2 / i / Solution

Write $p=1-R$ and, for a constant bank rate $s$, defineAssume $\sigma >0$, positive portfolio wealth, unrestricted stock fractions and the usual nonnegative-wealth admissibility. Once the rate has changed, the model is the Merton consumption-investment problem with $s=r$. Put ${\gamma }_0=\gamma (r)$. In its finite-value regime ${\gamma }_0>0$, the answer isHere ${\pi }_0^{\ast }$ is the fraction of portfolio wealth in the stock, so the dollar holding is ${\pi }_0^{\ast }{w}_t$.

The positive-coefficient condition matters: the stated signs of $\rho$ and $R$ alone do not guarantee finite infinite-horizon expected utility maximization. If ${\gamma }_0\le 0$, the unconstrained Merton consumption-investment problem has value $+\infty$ for $0<R<1$ and $-\infty$ for $R>1$. The remaining calculation therefore uses ${\gamma }_0>0$.