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The Inada conditions specify infinite marginal utility at zero consumption or wealth and zero marginal utility at infinity. For an increasing strictly concave utility function, they make the inverse marginal utility map cover all positive shadow prices, facilitating the state-price budget constraint method.

Part (c) gives $\mathbb{E}{M}_t\le 0$. Since $Y_t{X}_t\ge 0$, its integrated identity impliesfor every finite $t$. The cumulative consumption increases with $t$, so the monotone convergence theorem yieldsThis is the infinite-horizon state-price budget constraint. No terminal-wealth convergence or vanishing assumption is needed for this inequality.

Let $F(y)$ be the manager's utility function as a function of nonnegative terminal fund wealth. WriteThenBoth pieces are increasing and strictly concave functions. However, the derivative jumps upward from $a{e}^{-a{w}_0}$ to $b{e}^{-a{w}_0}$ at $w_0$. The payoff utility is increasing but not concave. A concave function must have nonincreasing one-sided slopes.

For concavification of incentive utility, the common tangent for exponential incentive utility replaces the upward kink by its common tangent. There are contact points $\ell <w_0<h$ and a common slope $s>0$ satisfyingSince $F(\ell )=-s/a$ and $F(h)=-s/b$, the chord condition gives $h-\ell =1/a-1/b$. Solving the derivative matching yields the contact pointsThe assumption that $w_0$ is large enough means, explicitly, thatso $\ell >0$. The inequalities $\ell <w_0<h$ follow from $\log q<q-1$ and $\log q>1-1/q$ for $q>1$.

The least concave majorant on $[0,\infty )$ isIt is increasing and concave: the derivative is decreasing outside the interval and equals $s$ inside. The tangent line lies above each original branch. Any concave majorant must lie above the chord joining the two contacts, so this one is the least.

The state-price budget constraint now reduces the problem to pointwise maximization of $\bar{U}(y)-qy$, where $q=\eta {\zeta }_T$ and $\eta >0$ is chosen so that $\mathbb{E}[{\zeta }_T{Y}^{\ast }]=w_0$. Because ${\bar{U}}^{\prime }(0)=a$ is finite, the nonnegative wealth constraint must be included. The optimal terminal wealth isAt $q=s$, any point of $[\ell ,h]$ maximizes the concavified objective. If $\kappa \ne 0$, the state-price density has a continuous log-normal distribution, so this event has probability zero. The optimizer then avoids $(\ell ,h)$ almost surely and satisfies $\bar{U}(Y^{\ast })=F(Y^{\ast })$. The supporting-line proof from part (i), with supergradients at the contacts, proves optimality for the concavified problem; the pointwise equality proves the same optimizer and value solve the original manager's problem. Nonnegative replication is available in the complete market.

If $\kappa =0$, the state-price density is deterministic. If the required deterministic mean terminal wealth lies in the linear segment, randomize between $\ell$ and $h$ at the common slope instead of choosing an interior wealth. This maintains the budget and attains the same concavified value. The Brownian market can replicate that bounded lottery even when its market price of risk is zero.

In the special case $r=\mu =0$, ${\zeta }_T=1$ and the fund has a fair-game wealth process. Since $\ell <w_0<h$, the manager uses fair-game gambling induced by an incentive fee:For example choose a threshold event in $W_T$ with the displayed probability and replicate its payoff. Then $w_t=\mathbb{E}[Y^{\ast }\mid {\mathcal{F}}_t]$ remains between $\ell$ and $h$, so the strategy respects nonnegative wealth. It earns no risk premium but raises expected incentive utility function above the value at the unrandomized $w_0$, because the common tangent lies strictly above the kink.

Assume $\sigma \ne 0$ and the usual Brownian filtration, so the single-risky-asset market is a complete market. With market price of risk $\kappa =(\mu -r)/\sigma$, the normalized state-price density isThe Itô formula shows that ${\zeta }_t{w}_t$ is a local martingale for a self-financing portfolio. For nonnegative admissible wealth it is a supermartingale, yielding the state-price budget constraintEvery integrable nonnegative terminal claim with equality is attainable in the complete market, by the Brownian martingale representation theorem. Its wealth process is $w_t={\zeta }_t^{-1}\mathbb{E}[{\zeta }_{T}X\mid {\mathcal{F}}_t]\ge 0$.

The Inada conditions $u^{\prime }(0+)=\infty$ and $u^{\prime }(\infty )=0$, together with strict concavity, make $I=(u^{\prime }{)}^{-1}$ a decreasing map from $(0,\infty )$ onto $(0,\infty )$. Pointwise optimization of $u(x)-y{\zeta }_{T}x$ gives the optimal terminal wealthwith scalar multiplier $y>0$. The question assumes a multiplier giving the required budget; utility expectations must also be well defined.

For any admissible terminal wealth $X$, the supporting-line inequality for a concave function givesTaking expectations and using the state-price budget constraint proves optimality:Strict concavity makes the optimal terminal wealth unique up to almost-sure equality.

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.