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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.

The PDF prints $w_0=0$ in the conditioning of the value function. Taken literally, nonnegative wealth and the state-price budget constraint force both wealth and consumption to remain zero, and the utility for $R>1$ has value $-\infty$. The meaningful value function underlying the subsequent requests uses $w_0=w>0$; the following calculation makes that source correction explicit.

Write $h_t={\bar{c}}_t>0$. Differentiating the exponentially weighted consumption habit gives the habit-state dynamicsMultiplying initial wealth, initial habit, investments, and consumption by $a>0$ multiplies the wealth and habit paths by $a$. The ratio $c/h$ is unchanged, while $U(ac)=a^{1-R}U(c)$. Thus the homogeneous value isThis is a multiplicative habit utility model: higher habit makes utility more negative at fixed consumption.

Set $S=R+\alpha >1$, $K=(S-1)/(R-1)>0$, and $\kappa =(\mu -r)/\sigma$. The instantaneous reward isFor smooth increasing, strictly concave wealth value, the Hamilton-Jacobi-Bellman equation isThe effective consumption shadow price with habit changes from $V_w$ to $D=V_w-\lambda V_h$. For $D>0$, consumption maximization givesFor $D<0$ the supremum is infinite, and for $D=0$ its zero supremum is approached only as consumption tends to infinity; a finite interior optimum therefore requires $D>0$. The portfolio maximum is $-{\kappa }^2{V}_w^2/(2V_{ww})$.

Let $x=w/h$, $z=v^{\prime }(x)$, and defineThe homogeneity derivatives areConsequently $D=h^{-R}d(x)$, and the reward conjugate is $h^{1-R}H(d(x))$. The reduced habit equation isFor completeness its feedback controls are $c^{\ast }/h=(K/d{)}^{1/S}$ and ${\theta }^{\ast }/h=-(\mu -r)v^{\prime }/({\sigma }^2{v}^{\prime \prime })$.

The wealth-variable Legendre dual $J(z)=v(x)-xz$ satisfies $J^{\prime }=-x$, $J^{\prime \prime }=-1/v^{\prime \prime }$, and $v=J-z{J}^{\prime }$. The effective shadow price becomesTherefore the dual equation for multiplicative habit investment iswith $\mathcal{D}>0$. The dependence of the reward conjugate on $J$ and $J^{\prime }$ is the remaining nonlinearity.

When $\lambda =0$, habit is fixed and the equation becomes a linear Euler equationThe forcing is a pure power $z^{1-1/S}$. Thus the Euler differential equation method gives a power particular solution plus the two homogeneous characteristic powers in the nondegenerate case. Economically, this is the Merton consumption-investment problem with effective relative risk aversion $S=R+\alpha$ and a constant reward multiplier. Writingits value and controls areIndeed $H(z)/{\delta }_S$ solves the dual equation, since its characteristic polynomial at $1-1/S$ equals $-{\delta }_S$. Unlike the $\lambda >0$ case, the fixed habit creates no feedback coupling between the dual value and the shadow price of consumption.