Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 40 / 1

Put $\kappa =(\mu -r)/\sigma$, assuming $\sigma >0$, and letThis state-price density prices the bank account and the risky asset. The Itô formula givesNonnegative wealth and consumption make the deflated cumulative gains a nonnegative local martingale, hence a supermartingale. Every admissible consumption stream therefore satisfies the state-price budget constraint $\mathbb{E}{\int }_0^{\infty }{\xi }_t{c}_{t}dt\le w_0$.

Write $I=(U^{\prime }{)}^{-1}$ for the inverse marginal utility. The Inada conditions make $I$ decreasing from infinity to zero. A positive Lagrange multiplier $y$ for the budget gives the pointwise condition $e^{-\rho t}{U}^{\prime }(c_t)=y{\xi }_t$. Thus, whenever a finite optimum exists and a multiplier exhausts the budget,The concave supporting-tangent inequality, integrated against the budget, proves optimality. These curvature and endpoint assumptions alone do not ensure that the infinite-horizon value is finite; part (b) gives the exact additional condition for power utility.

The optimal wealth is the conditional price of remaining consumption:It is nonnegative. Apply the Martingale representation theorem to the integrable total discounted consumption claim:Comparing $M_t={\xi }_t{w}_t^{\ast }+{\int }_0^t{\xi }_s{c}_s^{\ast }ds$ with the deflated wealth equation yieldsThis constructs the portfolio without assuming extra smoothness of the value function. If the stationary value $V$ is twice differentiable with $V^{\prime \prime }<0$, optimization of its Hamilton-Jacobi-Bellman equation instead gives ${\theta }^{\ast }(w)=-(\mu -r)V^{\prime }(w)/({\sigma }^2{V}^{\prime \prime }(w))$ and $c^{\ast }(w)=I(V^{\prime }(w))$.

For constant relative risk aversion utility, $I(z)=z^{-1/R}$. Define the Merton consumption constantThe Gaussian exponential moment givesWhen ${\gamma }_M>0$, the budget is $w_0=y^{-1/R}/{\gamma }_M$. Conditional pricing then gives $c_t^{\ast }={\gamma }_M{w}_t^{\ast }$, and the Merton consumption-investment problem hasThe resulting wealth is a geometric Brownian motion, so positive initial wealth remains positive. Substitution in the Hamilton-Jacobi-Bellman equation verifies the first-order conditions and the value; the budget argument in part (a) supplies global optimality.

The problem has a finite value precisely when ${\gamma }_M>0$. For $0<R<1$ and ${\gamma }_M\le 0$, the supremum is $+\infty$. Indeed, use the displayed risky fraction and consumption $c_t=k{w}_t$, $k>0$. Its expected utility isFor ${\gamma }_M<0$, sufficiently small $k$ already gives infinite utility. For ${\gamma }_M=0$, the finite values tend to infinity as $k\downarrow 0$.

For $R>1$ and ${\gamma }_M\le 0$, every admissible stream has utility $-\infty$. To see this, put $q=(R-1)/R$ and apply Holder inequality on $[0,T]\times \Omega$:The left side diverges as $T\to \infty$, forcing the positive utility-cost integral to diverge. Thus there is no finite-value optimization problem in either ill-posed case.