An investor has a (possibly negative) bank balance $x(t)$ at time $t$. For given positive $x(0), T, \mu, A$ and $r$, he wishes to choose his spending rate $u(t) \geqslant 0$ so as to maximize

$$\Phi(u ; \mu) \equiv \int_{0}^{T} e^{-\beta t} \log u(t) d t+\mu e^{-\beta T} x(T),$$

where $d x(t) / d t=A+r x(t)-u(t)$. Find the investor's optimal choice of control $u(t)=u_{*}(t ; \mu)$.

Let $x_{*}(t ; \mu)$ denote the optimally-controlled bank balance. By considering next how $x_{*}(T ; \mu)$ depends on $\mu$, show that there is a unique positive $\mu_{*}$ such that $x_{*}\left(T ; \mu_{*}\right)=0$. If the original problem is modified by setting $\mu=0$, but requiring that $x(T) \geqslant 0$, show that the optimal control for this modified problem is $u(t)=u_{*}\left(t ; \mu_{*}\right)$.