= Solution
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 \b[the habit-state dynamics]
$$
\boxed{dh_t=\lambda(c_t-h_t)\,dt.}
$$
Multiplying 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 \b[the homogeneous value is]
$$
\boxed{V(w,h)=h^{1-R}v(w/h).}
$$
This 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 is
$$
g(c,h)=\frac{h^\alpha c^{1-S}}{1-R}
=K h^\alpha\frac{c^{1-S}}{1-S}.
$$
For smooth increasing, strictly <concave> wealth value, the <Hamilton-Jacobi-Bellman equation> is
$$
0=-\rho V+rwV_w-\lambda hV_h
+\sup_{\theta\in\mathbb R}
\left\{\tfrac12\sigma^2\theta^2V_{ww}+(\mu-r)\theta V_w\right\}
+\sup_{c>0}\{g(c,h)-c(V_w-\lambda V_h)\}.
$$
The <effective consumption shadow price with habit> changes from $V_w$ to $D=V_w-\lambda V_h$. For $D>0$, consumption maximization gives
$$
c^*=(Kh^\alpha/D)^{1/S},\qquad
\sup_{c>0}\{g(c,h)-Dc\}
=-\frac S{S-1}(Kh^\alpha)^{1/S}D^{1-1/S}.
$$
For $D<0$ 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 $-\kappa^2V_w^2/(2V_{ww})$.
Let $x=w/h$, $z=v'(x)$, and define
$$
d(x)=(1+\lambda x)v'(x)-\lambda(1-R)v(x),\qquad
H(d)=-\frac S{S-1}K^{1/S}d^{1-1/S}.
$$
The <homogeneity> derivatives are
$$
V_w=h^{-R}v',\quad V_{ww}=h^{-R-1}v'',\quad
V_h=h^{-R}[(1-R)v-xv'].
$$
Consequently $D=h^{-R}d(x)$, and the reward conjugate is $h^{1-R}H(d(x))$. \b[The reduced habit equation is]
$$
\boxed{(r+\lambda)xv'-[\rho+\lambda(1-R)]v
-\tfrac12\kappa^2\frac{(v')^2}{v''}+H(d(x))=0.}
$$
For completeness its feedback controls are $c^*/h=(K/d)^{1/S}$ and $\theta^*/h=-(\mu-r)v'/(\sigma^2v'')$.
The <wealth-variable Legendre dual> $J(z)=v(x)-xz$ satisfies $J'=-x$, $J''=-1/v''$, and $v=J-zJ'$. The effective shadow price becomes
$$
\mathcal D(z)=z-\lambda RzJ'(z)+\lambda(R-1)J(z).
$$
Therefore \b[the <dual equation for multiplicative habit investment> is]
$$
\boxed{\tfrac12\kappa^2z^2J''
+(\rho-r-\lambda R)zJ'
-[\rho+\lambda(1-R)]J
-\frac S{S-1}K^{1/S}
\left[z-\lambda RzJ'+\lambda(R-1)J\right]^{1-1/S}=0,}
$$
with $\mathcal D>0$. The dependence of the reward conjugate on $J$ and $J'$ is the remaining nonlinearity.
When $\lambda=0$, habit is fixed and the equation becomes \b[a linear Euler equation]
$$
\boxed{\tfrac12\kappa^2z^2J''+(\rho-r)zJ'-\rho J+H(z)=0.}
$$
The 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. Writing
$$
\delta_S=\frac{\rho-(1-S)[r+\kappa^2/(2S)]}{S}>0,
$$
its value and controls are
$$
\boxed{v(x)=\frac{K\delta_S^{-S}}{1-S}x^{1-S},\qquad
c^*=\delta_S w,\qquad \theta^*=\frac{\mu-r}{S\sigma^2}w.}
$$
Indeed $H(z)/\delta_S$ 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.
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