You cannot extract tomorrow's ETF weight from this paper. Mastrogiacomo and Tarsia say a closed form is unattainable and leave numerical schemes out of scope. They give necessary conditions for a present-biased investor whose recursive utility faces a second recursive-utility constraint. Applying those conditions means solving for the trading rule and two multipliers together. The sign of the cost multiplier remains unresolved in the general result.
Why each self needs an equilibrium
With nonexponential discounting, including a hyperbolic kernel, a plan chosen today can cease to be optimal tomorrow. Bellman's principle fails. Following Ekeland and Pirvu and their own unconstrained paper, the authors treat each instant's self as a player. They seek a subgame-perfect equilibrium, a feedback rule Φ(s,x) under which no self gains to first order from a deviation over [t, t+ε]. The investor's cost is y(t;t), the initial value of a BSDE defining recursive preferences. This class includes Uzawa-type utilities.
A second recursive utility supplies the constraint: its initial value J̄ must belong to a closed interval Γ. In Example 1, terminal wealth is the terminal condition and the driver defines a g-expectation. The constraint then caps a risk measure of terminal wealth; the entropic measure is the example the authors name.
Any constrained equilibrium, the theorem says, has two multipliers in [-1,1], one for the cost and one for the constraint. They obey ψ² + ψ̄² = 1, so both cannot vanish. If the constraint multiplier is nonzero, a transversality inequality puts J̄ at an endpoint of Γ. The Hamiltonian inequality ψδH + ψ̄δH̄ ≥ 0 must hold for every admissible control value at the start of each subgame. When the cost multiplier has absolute value 1, the constraint is inactive and the authors recover their earlier unconstrained condition.
The proof uses the penalty J_ϱ = {(J+ϱ)² + d²_Γ(J̄)}^{1/2}. Ekeland's principle then gives an approximate minimizer within √ϱ of the equilibrium, using a distance between controls based on how often they disagree. Since an equilibrium cost need not be a minimum, the penalty can vanish. The proof treats that possibility in two subcases. A local-uniqueness assumption (Assumption 2, ϱ in ]0, δ̄²[) rules it out.
There is no data. The illustration is a Merton-type investment-consumption problem: r, μ and σ are constant in ]0,∞[, and wealth stays in ]0,∞[. A discounted Uzawa-type recursive utility constrains the investor, with consumption weighted by a scale k in [0,∞[.
From continuous-time controls to an ETF
Our ETF implementation replaces the paper's single risky asset with a US equity ETF and puts the remaining wealth in cash. In the model, ζ is the fraction of wealth invested in the risky asset; c is consumption per unit of wealth. Both are continuous-time controls. We reset positions daily. Each day starts a new subgame, with t advancing, a set to that day's wealth, and the constraint applied afresh to that day's J̄.
One state variable and one Brownian motion suit one ETF. The main theorem requires a bounded control set, however, so this implementation needs a leverage cap on ζ and a cap on c. The paper otherwise allows consumption in [0,∞[. Capping it changes what can be inferred: the k = 0 positivity result relies on c↑∞, and its Section 4 sign conclusions therefore do not carry over as written. Only with both caps does the truncation corollary, which uses balls of radius ||ū||∞ + j, become unnecessary.
The constraint governs the initial value of the auxiliary utility for each (t,a). It supplies no running or pathwise stop.
Can the multiplier sign be fixed?
The authors say "no definitive information can be obtained" about the scalar multiplier's sign. They then say the departure from the classical Hamiltonian form "does not reduce their analytical content, as the equilibrium system remains fully characterized through the adjoint equations and the generalized Hamiltonian inequality."
For a trader, "fully characterized" still leaves ψ² + ψ̄² = 1 to solve alongside the rule. Under qualification assumptions in a classical constrained problem, the multiplier's sign is fixed and a single Lagrangian can be minimized. Here the ratio of the multipliers depends on adjoint values that are also unknown. A proposed rule must subsequently pass the transversality inequality and satisfy Γ. The authors' defence does not settle the work of finding those multipliers.
Their portfolio example gets further. At k = 0, consumption drops out of the constraint, making this the case closest to a pure terminal-wealth limit. Under additional assumptions taken from Theorem 3 of their earlier paper, the qualification condition holds and one multiplier is strictly positive. At k ≠ 0, the sign result is partial. It hinges on whether kῡ(Υ(ξ)) exceeds (−ξ̄)Υ(ξ); both expressions depend on adjoint values the paper leaves unsolved.
The k = 0 result merits space in the abstract. Readers currently find it in the last pages of Section 4.
The equations behind a weight
A position requires four linear adjoint BSDEs in addition to the state equation and two utility BSDEs. The cost has first- and second-order adjoints; so does the constraint. Their coefficients depend on the equilibrium path, turning the calculation into a fixed-point problem for the rule. The authors say a closed-form equilibrium strategy is unattainable and put numerical schemes outside their scope.
Once the adjoints are available, the Hamiltonian inequality is quadratic in ζ − ζ̄. Its leading coefficient is (a/2)σ²(ψΞ + ψ̄Ξ̄), which must be nonnegative. Hold c at its equilibrium value and the inequality becomes an equation linking ξ, ϑ, ξ̄, ϑ̄ and the multipliers to μ − σγ and μ − σγ̄. The analogous condition determines the weight in the authors' unconstrained paper. Here it contains multipliers that must also be found.
Even a computed rule remains a candidate. As the authors concede, "without an independent existence result, the maximum principle may remain purely formal." Satisfying every stated condition would still leave open whether a numerical rule is an equilibrium.
What the backtest must supply
A backtest here is our own construction. The paper leaves nearly every input it would require for us to choose.
- Risk limit. Γ is introduced as a closed interval, though Section 4 calls it "a general closed subset of the Euclidean space", at odds with Definition 2.
- Discounting. The functions ℏ are described only as "regular" on [t, T].
- Costs and estimation. The model is frictionless, with fixed r, μ and σ in ]0,∞[. We must choose how to handle daily turnover costs and estimation error in μ.
- Benchmarks. The authors name the unconstrained recursive equilibrium from their earlier paper and the time-consistent optimum under exponential discounting.
The C² requirement on coefficients excludes Heston-type volatility, as the authors acknowledge. It matters little for a single ETF with constant σ.
We are running this test now. For now, the paper establishes necessary conditions for a decision rule; our backtest must supply the rest. A structural condition fixing the sign of the cost multiplier ψ would change how much weight this framework deserves. The authors list monotonicity of the BSDE driver in the utility variable as one open possibility. It would bring the multipliers into a single Hamiltonian, while leaving existence to be proved separately.