Admissibility and Transversality Checks in Merton Problems

Why a Formal HJB Optimizer Needs Further Checks

Solving the Merton Hamilton-Jacobi-Bellman equation produces feedback formulas, but those formulas are not automatically feasible or optimal. The risky-asset position may violate a portfolio constraint, the consumption rate may fail to be positive, or the discounted utility and terminal value may not satisfy the limits needed for an infinite-horizon verification argument. These are separate checks: a smooth solution of the HJB equation alone does not establish that its optimizer is an admissible policy.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a standard Brownian motion $B_t$. Consider wealth $X_t$, dollar investment $\pi_t$ in a risky asset, and consumption rate $c_t$, governed by

dX_t=\big(rX_t+(\mu-r)\pi_t-c_t\big)\,dt+\sigma\pi_t\,dB_t,\qquad X_0=x>0,

where $r$ is the constant interest rate, $\mu$ the risky asset’s expected return, and $\sigma>0$ its volatility. The coefficient of $dt$ is the wealth drift; $\sigma\pi_t$ is the diffusion coefficient. On each finite interval $[0,T]$, controls are progressively measurable and satisfy $\mathbb{E}\int_0^T\pi_t^2\,dt<\infty$ and $\mathbb{E}\int_0^T c_t\,dt<\infty$. We require $X_t\ge0$ and $c_t\ge0$, with $c_t>0$ almost everywhere when using CRRA utility with exponent below zero. These conditions describe the admissibility checks used here; a particular problem may impose additional constraints.

For risk aversion $\gamma>0$, $\gamma\ne1$, let $U(c)=c^{1-\gamma}/(1-\gamma)$. The infinite-horizon objective is $\mathbb{E}\int_0^\infty e^{-\rho t}U(c_t)\,dt$, with discount rate $\rho>0$. Write $p=1-\gamma$, the risky-asset premium as $\mu-r$, and define

q=r+\frac{(\mu-r)^2}{2\gamma\sigma^2}, \qquad \alpha_*=\frac{\mu-r}{\gamma\sigma^2}, \qquad \kappa_*=\frac{\rho-(1-\gamma)q}{\gamma}.

The unconstrained CRRA HJB candidate has investment $\pi_t=\alpha_*X_t$ and consumption $c_t=\kappa_*X_t$. The fraction $\alpha_*$ and rate $\kappa_*$ are candidates, not yet an admissibility or optimality conclusion. In particular, a positive consumption rate requires $\kappa_*>0$.

Wealth, Integrability, and the Terminal Check

Proposition: Suppose $\kappa_*>0$ and use the feedback policy $\pi_t=\alpha_*X_t,\ c_t=\kappa_*X_t$. Then wealth is strictly positive at every finite time, the controls satisfy the stated finite-horizon integrability conditions, and $$ \mathbb{E}\big[e^{-\rho t}X_t^{1-\gamma}\big] =x^{1-\gamma}e^{-\kappa_*t}. $$ Consequently, the discounted utility integral is finite in absolute expectation and the candidate’s transversality term $\mathbb{E}[e^{-\rho T}V(X_T)]$ tends to zero for $V(x)=x^{1-\gamma}\kappa_*^{-\gamma}/(1-\gamma)$.

Proof. Under the feedback policy, the wealth equation becomes $dX_t=mX_t\,dt+vX_t\,dB_t$, where $m=r+(\mu-r)\alpha_*-\kappa_*$ and $v=\sigma\alpha_*$. Its explicit solution is $X_t=x\exp((m-\tfrac12v^2)t+vB_t)$, which is strictly positive. To verify this expression, set $f(t,z)=x\exp((m-\tfrac12v^2)t+vz)$. Its derivatives are $f_t=(m-\tfrac12v^2)f$, $f_z=vf$, and $f_{zz}=v^2f$. Ito’s formula gives $df=(f_t+\tfrac12f_{zz})dt+f_z\,dB_t=mf\,dt+vf\,dB_t$, including the correction $\tfrac12v^2f\,dt$. Thus it solves the stated equation with initial value $x$.

The normal moment-generating formula gives $\mathbb{E}[X_t^a]=x^a\exp((am+\tfrac12a(a-1)v^2)t)$ for every real $a$. For $a=1$ and $a=2$, these moments are finite and bounded on each finite time interval. Hence $\mathbb{E}\int_0^T\pi_t^2dt=\alpha_*^2\int_0^T\mathbb{E}[X_t^2]dt<\infty$, and $\mathbb{E}\int_0^Tc_tdt=\kappa_*\int_0^T\mathbb{E}[X_t]dt<\infty$.

For $a=p=1-\gamma$, use $\alpha_*=(\mu-r)/(\gamma\sigma^2)$. Direct substitution gives $m-\tfrac12\gamma v^2=q-\kappa_*$. Therefore $pm+\tfrac12p(p-1)v^2=p(q-\kappa_*)$. Since $\rho=pq+\gamma\kappa_*$, it follows that $pm+\tfrac12p(p-1)v^2-\rho=-\kappa_*$, proving the displayed moment identity. Moreover, $$ \mathbb{E}\int_0^\infty e^{-\rho t}|U(c_t)|\,dt =\frac{\kappa_*^{\,1-\gamma}x^{1-\gamma}}{|1-\gamma|} \int_0^\infty e^{-\kappa_*t}\,dt =\frac{\kappa_*^{-\gamma}x^{1-\gamma}}{|1-\gamma|}<\infty. $$ Finally, the terminal expectation for the displayed $V$ is a constant multiple of $e^{-\kappa_*T}$, which tends to zero because $\kappa_*>0$. $\square$

The proposition checks the candidate under the physical measure $\mathbb{P}$. The transversality statement is an expectation limit for this policy; it is not a claim that every admissible competing policy has the same terminal behavior. An infinite-horizon verification argument must handle its own admissibility and limiting hypotheses, as well as the HJB inequality and equality along the candidate.

Worked Example: A Candidate That Passes the Infinite-Horizon Checks

Take $x=1,\ r=3/100,\ \mu-r=4/100,\ \sigma=1/5,\ \gamma=2,\ \rho=1/10$. Compute the candidate rates, expected discounted utility, and terminal limit.

First, $\gamma\sigma^2=2(1/25)=2/25$, so $\alpha_*=(1/25)/(2/25)=1/2$. Also, $2\gamma\sigma^2=4/25$, hence $(\mu-r)^2/(2\gamma\sigma^2)=(1/625)/(4/25)=1/100$, and $q=3/100+1/100=1/25$. Thus $$ \kappa_*=\frac{1/10-(1-2)(1/25)}{2} =\frac{1/10+1/25}{2} =\frac{7}{100}>0. $$

The candidate is $\pi_t=X_t/2,\ c_t=7X_t/100$. Its utility exponent is $p=-1$. By the proposition, $\mathbb{E}[e^{-\rho t}X_t^{-1}]=e^{-7t/100}$. Therefore $$ \mathbb{E}\int_0^\infty e^{-t/10}U(c_t)\,dt =-\frac{100}{7}\int_0^\infty e^{-7t/100}\,dt =-\frac{100}{7}\cdot\frac{100}{7} =-\frac{10000}{49}. $$

Here $V(x)=-\frac{10000}{49}x^{-1}$, so $\mathbb{E}[e^{-T/10}V(X_T)]=-\frac{10000}{49}e^{-7T/100}\to0$. This checks the terminal condition and matches the calculated objective at $x=1$. It does not replace the HJB verification inequality for other admissible policies.

Portfolio Constraints and Failed Formal Policies

Suppose the investor is required to keep the risky investment between zero and wealth: $0\le\pi_t\le X_t$. Under a proportional rule this is exactly $0\le\alpha\le1$. The unconstrained formula $\alpha_*$ is feasible only if it lies in that interval. More generally, with a closed interval constraint $\alpha\in[\underline{\alpha},\overline{\alpha}]$, the quadratic portfolio term in the HJB is maximized at the unconstrained point if feasible, and otherwise at the nearest endpoint. This follows because the term $\alpha(\mu-r)-\tfrac12\gamma\sigma^2\alpha^2$ is strictly concave: its derivative is $(\mu-r)-\gamma\sigma^2\alpha$, and its unique unconstrained maximum is $\alpha_*$.

Worked Example: The Interior Portfolio Formula Violates a Constraint

Let $r=1/20,\ \mu-r=1/10,\ \sigma=1/5,\ \gamma=2,\ \rho=1/10$, and impose $0\le\pi_t\le X_t$. The unconstrained investment fraction is $$ \alpha_*=\frac{1/10}{2(1/25)}=\frac{5}{4}>1. $$

It violates the upper portfolio constraint. To see which constrained fraction maximizes the portfolio term, its derivative is $1/10-(2/25)\alpha$. For $0\le\alpha\le1$, this derivative is at least $1/10-2/25=1/50>0$; hence the constrained maximum is at $\alpha=1$, not $5/4$.

At that boundary, the maximized portfolio contribution is $(\mu-r)\alpha-\tfrac12\gamma\sigma^2\alpha^2=1/10-1/25=3/50$. Thus the corresponding effective return term is $q_{\mathrm{con}}=r+3/50=11/100$, and the consumption rate obtained from the HJB coefficient is $$ \kappa_{\mathrm{con}}= \frac{\rho-(1-\gamma)q_{\mathrm{con}}}{\gamma} =\frac{1/10+11/100}{2} =\frac{21}{200}>0. $$

Check: the proposed constrained feedback $\pi_t=X_t,\ c_t=21X_t/200$ satisfies both portfolio inequalities, since $0\le X_t\le X_t$, and has positive consumption. Substituting $\alpha=1$ into the derivative gives $1/50>0$, confirming that moving left from the boundary decreases the portfolio objective. The unconstrained formula fails the constraint even though its own formal consumption coefficient is positive: direct substitution gives $q=1/20+1/16=9/80$ and $\kappa_*=(1/10+9/80)/2=17/160>0$.

Worked Example: A Nonpositive Consumption Rate

Take $\gamma=1/2,\ r=1/50,\ \mu-r=1/10,\ \sigma=1/5,\ \rho=1/10$. The risk contribution is $$ \frac{(\mu-r)^2}{2\gamma\sigma^2} =\frac{1/100}{(1)(1/25)} =\frac{1}{4}, \qquad q=\frac{1}{50}+\frac14=\frac{27}{100}. $$

Since $1-\gamma=1/2$, the formal consumption rate is $$ \kappa_*= \frac{1/10-(1/2)(27/100)}{1/2} =\frac{20/200-27/200}{1/2} =-\frac{7}{100}. $$

The feedback $c_t=\kappa_*X_t$ is negative whenever $X_t>0$, so it violates $c_t\ge0$. The formal HJB first-order condition therefore does not produce an admissible consumption policy in this parameter regime. The conclusion is not that negative consumption should be implemented; rather, the proposed interior formula has failed its feasibility test, and one must analyze the constrained optimization problem and its value separately.

What Transversality Does—and Does Not—Establish

For the CRRA candidate, $\kappa_*>0$ gives a particularly transparent check: the discounted $p$-th wealth moment decays as $e^{-\kappa_*t}$. The same decay makes both the candidate’s absolute expected utility integral finite and its candidate terminal value vanish. If $\kappa_*=0$, the displayed integral has no exponential decay and the candidate’s consumption coefficient is not strictly positive. If $\kappa_*<0$, the proportional feedback prescribes negative consumption. In either case, the preceding verification cannot be applied by simply retaining the formal formula.

These calculations are also why finite-horizon admissibility and infinite-horizon verification should not be conflated. The candidate has finite expected squared investment on every bounded interval because its lognormal wealth has finite second moments there. That fact alone does not establish integrability over $[0,\infty)$, nor does it imply a terminal limit. The infinite-horizon checks use the discounted moment explicitly.

Common mistake. A frequent error is to compute $\alpha_*$ and $\kappa_*$, observe that the HJB first-order conditions hold, and call the resulting policy optimal without checking the constraints. In the portfolio example, that calculation yields $\alpha_*=5/4$, but the actual constraint requires $\alpha\le1$. The correction is to test $\alpha_*$ against the feasible set first; here the objective is increasing throughout $[0,1]$, so the admissible maximizing fraction is $1$. A separate check of $\kappa_*>0$, finite expected utility, and the terminal expectation is still required.

Takeaway: For a proportional Merton candidate, check positivity of wealth and consumption, finite-horizon control integrability, feasibility of the risky fraction, finiteness of discounted expected utility, and the terminal expectation. Passing these tests makes the candidate eligible for verification; it does not by itself prove the HJB inequality for competing controls.

Check Your Understanding

For each question, state any parameter or positivity assumptions needed before applying the calculation.

For given $r,\mu,\sigma,\gamma,\rho$, compute $\alpha_*$, $q$, and $\kappa_*$. Which inequality makes proportional consumption strictly positive?
If the constraint is $0\le\pi_t\le X_t$, what condition on $\alpha_*$ makes the unconstrained investment fraction feasible?
Using the wealth moment formula, show why the discounted $1-\gamma$ moment of the candidate decays at rate $\kappa_*$.
Why does finite expected squared investment on every bounded time interval not by itself establish an infinite-horizon verification condition?
In a regime with $\kappa_*\le0$, identify which admissibility check fails for the proposed feedback $c_t=\kappa_*X_t$.

Admissibility and Transversality Checks in Merton Problems

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