 Solution

Use dollar holdings ${\theta }_0,\dots ,{\theta }_n$, including the traded index, and let $m=\log M$. The Itô formula givesDefine the state-dependent correlation $q(m)=\rho (e^m)$, $d(m)=\sqrt{1-q(m{)}^2}>0$, and constants $p_i=({\mu }_i-r)/{\sigma }_i$. The self-financing portfolio with consumption has wealth equationSince all these assets are traded and $d(m)>0$, the volatility map is invertible. It is useful to optimize over Brownian portfolio exposureswhose market price of risk vector isIndeed ${\sum }_i({\mu }_i-r){\theta }_i=\ell \cdot y$. Put $L^2(m)=p_0^2+{\sum }_{i=1}^n{\ell }_i(m{)}^2$.

For $R\ne 1$, normalize constant relative risk aversion utility to $U(c)=c^{1-R}/(1-R)$. Scaling wealth, holdings, and consumption by $a>0$ leaves the index state unchanged and multiplies reward by $a^{1-R}$. ThusFor a finite smooth value with $V_{ww}<0$, the Hamilton-Jacobi-Bellman equation, including the shared-noise cross derivative, isThe last term is essential: index changes and the $W^0$ component of wealth are correlated. The first-order conditions yieldBecause $V_w=w^{-R}f$, $V_{ww}=-R{w}^{-R-1}f$, and $V_{wm}=w^{-R}{f}^{\prime }$, the optimized portfolio contribution isSubstitution gives the requested second-order nonlinear equationRecovering dollar holdings from the exposures gives the optimal controlsOnly the index holding carries the extra intertemporal hedging demand, because the index is the source of state variation. The stock positions hedge their common exposure through the index.

The power transformation of a complete-market investment equation gives a useful further simplification. Let $g=f^{1/R}$, and defineExpanding the squared term and substituting $f=g^R$ cancels the two $(g^{\prime }{)}^2$ terms. The power-transformed investment equation is linear:Then $c^{\ast }=w/g$ and $f^{\prime }/f=R{g}^{\prime }/g$. A practical finite difference method solves this linear differential equation on an expanding truncated interval in $m$, enforcing the economically relevant positive solution and checking domain and mesh convergence. If correlation approaches limits strictly inside $(-1,1)$ and the corresponding ${\delta }_{\pm }>0$, the constant-coefficient Merton consumption-investment problem gives endpoint approximations $g\to 1/{\delta }_{\pm }$. Without such asymptotics one must determine appropriate growth/transversality conditions; arbitrary fixed endpoint values are not justified. The statement's smooth decreasing correlation alone does not ensure globally bounded market prices of risk or a finite value. Any numerical candidate must also satisfy admissibility and the investment value transversality condition.

The printed $R>0$ includes $R=1$. In that case choose $U(c)=\log c$, whose scaling is additive:There is no shared-noise cross derivative because $V_{wm}=0$. The logarithmic case isThus the dollar holdings are the preceding formulas with $R=1$ and the $f^{\prime }/f$ hedge term omitted. This linear differential equation can be solved by the same truncation and convergence strategy.