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Concavification replaces a nonconcave incentive utility function by its least concave majorant. In a complete market with an atomless state-price density, the optimizing payoff may avoid all intervals where the majorant strictly exceeds the original utility, making the relaxed optimum exactly attainable. At a pricing atom corresponding to a linear segment, a lottery between contact points can attain the same value at unchanged cost.

A time-homogeneous Markov decision process specifies a state space, admissible actions at each state, an action-dependent transition kernel and one-step reward, with neither kernel nor reward depending on time. A policy chooses actions using the observed history. The positive programming case has nonnegative rewards and maximizes expected undiscounted total reward, allowing infinite values; its finite-horizon values increase to the infinite-horizon supremum under the appropriate approximation argument.

Here let ${\tau }_B$ be the absorption time of the underlying simple random walk at $0,N$. Every admissible stopping time $T$ is at most ${\tau }_B$. The reward is bounded by $R_{\max }={\max }_{j}r(j)$, and $E_i{\tau }_B=i(N-i)$, as follows by solving $h_i=1+(h_{i-1}+h_{i+1})/2$ with zero endpoint values. In particular absorption is almost surely finite. Truncating any policy at $T\wedge n$ changes its reward expectation by at most $2R_{\max }{P}_i({\tau }_B>n)\le 2R_{\max }i(N-i)/n$. Since the truncated policy is allowed in the $n$-step problem,This proves $V_n\to V$, even uniformly on the finite state space.

The finite-horizon recursion starts with $V_0=r$, fixes the endpoint rewards and hasTaking limits shows $V\ge r$ and $2V(i)\ge V(i-1)+V(i+1)$, so $V$ is discretely concave. If $W$ is any concave function majorizing $r$, induction in this same recursion gives $V_n\le W$, and hence $V\le W$. Therefore $V$ is the smallest concave majorant of the rewards. At a point where $V(i)>r(i)$ the recursion forces equality with the neighboring average; the value graph is linear through each consecutive continuation region.

An optimal policy stops upon first entering the contact set $\{i:V(i)=r(i)\}$ and continues elsewhere. Before stopping the neighboring-average equality makes $V$ of the walk a bounded martingale; the stopping time is bounded by ${\tau }_B$. Applying the bounded stopping identity by first truncating the time and then passing to the limit proves the expected terminal reward equals $V(i)$.

Distinct rewards do not guarantee uniqueness. With $N=2$ and $r(0)=0,r(1)=1,r(2)=2$, the reward function is already linear and $V=r$. From state one, stopping gives one, while continuing to the next step also gives expected reward $(0+2)/2=1$. Both policies are optimal. Thus the claimed uniqueness is false even when all rewards are distinct.

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.