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arxiv: 2606.02945 · v1 · pith:PS2QIQGBnew · submitted 2026-06-01 · 💱 q-fin.MF · math.OC· q-fin.PM

Infinite Horizon Optimal Consumption: Intertemporal Hedging under Epstein-Zin Preferences

Erhan Bayraktar , Emmet Lawless This is my paper

Pith reviewed 2026-06-28 11:17 UTC · model grok-4.3

classification 💱 q-fin.MF math.OCq-fin.PM
keywords optimal consumptionEpstein-Zin utilityvariational characterisationincomplete marketsHamilton-Jacobi-Bellman equationintertemporal hedgingstochastic differential utilityverification theorem
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The pith

The value function in this infinite-horizon Epstein-Zin consumption-investment problem is the unique minimiser of a functional whose Euler-Lagrange equation recovers the Hamilton-Jacobi-Bellman equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies optimal consumption and investment over an infinite horizon for an investor with Epstein-Zin stochastic differential utility in an incomplete market with stochastic opportunities. It separates risk aversion from intertemporal substitution and works in the regime where θ lies between 0 and 1, ensuring a unique generalised utility process exists for any admissible consumption stream. The central result is a variational characterisation: the value function is the unique minimiser of a functional, and this minimiser is strictly positive, bounded, and classical even if the functional is non-convex. A verification theorem then equates any such minimiser with the value function and supplies explicit feedback representations for the optimal policies. The argument proceeds by changing measure to the myopic probability, invoking uniqueness for the associated Epstein-Zin backward stochastic differential equations, and applying a perturbation argument.

Core claim

The value function is the unique minimiser of a functional whose Euler-Lagrange equation coincides with the Hamilton-Jacobi-Bellman equation. Although the functional may be non-convex, the direct method yields existence, every minimiser is strictly positive, bounded, and classical, and a verification theorem identifies any minimiser with the value function while giving feedback representations for optimal consumption and investment policies. The proof combines a change of measure to the myopic probability with uniqueness results for Epstein-Zin BSDEs and a perturbation argument for optimality.

What carries the argument

A variational functional whose Euler-Lagrange equation matches the Hamilton-Jacobi-Bellman equation, together with a change of measure to the myopic probability and uniqueness for Epstein-Zin BSDEs.

If this is right

  • Optimal consumption and investment policies admit explicit feedback representations derived from the minimiser.
  • Existence of the value function holds via the direct method even when the functional is non-convex.
  • The characterisation applies to concrete models with stochastic volatility, Gaussian excess returns, and fat-tailed excess returns.
  • Intertemporal hedging demand is captured through the variational problem in incomplete markets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The variational approach could be adapted to finite-horizon versions or to other recursive utility specifications that admit similar BSDE uniqueness.
  • Minimisation of the functional might serve as the basis for numerical approximation schemes that avoid direct solution of the nonlinear HJB equation.
  • The myopic-measure change and perturbation technique may transfer to other stochastic control problems whose value functions satisfy non-standard Hamilton-Jacobi-Bellman equations.

Load-bearing premise

Existence of a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams when θ lies in (0,1).

What would settle it

An explicit consumption-investment problem in the θ∈(0,1) regime whose associated functional possesses a minimiser that fails to satisfy the Hamilton-Jacobi-Bellman equation or whose verification theorem does not recover the true value function.

Figures

Figures reproduced from arXiv: 2606.02945 by Emmet Lawless, Erhan Bayraktar.

Figure 1
Figure 1. Figure 1: Optimal consumption ratio l ∗ (y) = g(y) −ν (solid lines) and the myopic bench￾mark κ(y) (dashed lines) as a function of the CIR state variable y for γ = 2 (left panel) and γ = 4 (right panel). Parameters: r0 = 0.05, r1 = 0, σ = 1, β = 0.08, λ = 0.47, b0 = 5, b1 = 0.0225, a = 0.25, ρ = −0.5 taken from (Xing, 2017). 5.1.2 Kim-Omberg excess returns Next, we present the classical model of Gaussian excess retu… view at source ↗
Figure 2
Figure 2. Figure 2: Optimal consumption ratio l ∗ (y) = g(y) −ν (left panel) and optimal portfolio weight π ∗ (y) (right panel) as a function of the state variable y (within a 99% confidence interval for the long-run stationary density of Y ), for (γ, δ) = (2, 9) (dotted lines), (γ, δ) = (2, 5) (dashed lines) and (γ, δ) = (4, 5) (solid lines). Parameters: r = 0.02, β = 0.05, σ = 0.15, b = 0.50, µ¯ = 0.06, a = 0.05, ρ = −0.30.… view at source ↗
Figure 3
Figure 3. Figure 3: Optimal consumption ratio l ∗ (y) = g(y) −ν (left panel) and optimal portfolio weight π ∗ (y) (right panel) as a function of the state variable y (within a 99% confidence interval for the long-run stationary density of Y ), for (γ, δ) = (2, 9) (dotted lines), (γ, δ) = (2, 5) (dashed lines) and (γ, δ) = (4, 5) (solid lines). Parameters: r = 0.02, β = 0.05, σ = 0.15, b = 0.50, µ¯ = 0.06, a = 0.05, ρ = −0.30.… view at source ↗
read the original abstract

We study an infinite-horizon optimal consumption-investment problem for an investor with Epstein-Zin stochastic differential utility with stochastic investment opportunities in an incomplete market. Risk aversion and intertemporal substitution are separated, and we work in the regime $\theta\in(0,1)$, where there exists a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams. Our main contribution is a variational characterisation of the value function. We show that the value function is the unique minimiser of a functional whose Euler-Lagrange equation coincides with the Hamilton-Jacobi-Bellman equation. Although the functional may be non-convex, the direct method yields existence, and we prove every minimiser is strictly positive, bounded, and classical. A verification theorem identifies any minimiser with the value function and gives feedback representations for optimal consumption and investment policies. The proof combines a change of measure to the myopic probability with uniqueness results for Epstein-Zin BSDEs and a perturbation argument for optimality. Examples with stochastic volatility, Gaussian excess returns, and fat-tailed excess returns illustrate the scope of the framework and its implications for intertemporal hedging.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the infinite-horizon optimal consumption-investment problem under Epstein-Zin stochastic differential utility in an incomplete market with stochastic investment opportunities. It works in the regime θ∈(0,1) and claims that the value function is the unique minimizer of a (possibly non-convex) functional whose Euler-Lagrange equation recovers the HJB equation. Existence of a minimizer is obtained via the direct method; every minimizer is shown to be strictly positive, bounded and classical; and a verification theorem equates minimizers to the value function while supplying feedback representations for optimal consumption and investment. The proofs combine a myopic-measure change, external uniqueness results for Epstein-Zin BSDEs, and a perturbation argument.

Significance. If the foundational uniqueness assumption on generalized utility processes holds, the variational characterization supplies a new route to existence, regularity and verification that may be useful for problems where standard dynamic-programming arguments are intractable. The explicit examples with stochastic volatility, Gaussian returns and fat-tailed returns illustrate concrete implications for intertemporal hedging demands.

major comments (2)
  1. [Abstract] Abstract (regime statement): the entire variational problem, the definition of the objective functional via U^c, the perturbation argument, and the verification theorem presuppose that a unique generalized utility process exists for every non-negative progressively measurable consumption stream when θ∈(0,1). The manuscript invokes external uniqueness results for Epstein-Zin BSDEs but does not identify the precise theorem or set of conditions that guarantee uniqueness for arbitrary (including non-Markovian) streams; without this grounding the functional is not demonstrably well-defined on the whole admissible set.
  2. [Verification theorem] Verification theorem (perturbation argument): the abstract states that the functional may be non-convex yet the direct method still yields existence, and that a perturbation argument identifies minimizers with the value function. No details are supplied on how the perturbation is constructed to avoid post-hoc choices or to obtain the comparison principle needed for optimality; this step is load-bearing for the claim that every minimizer solves the original control problem.
minor comments (2)
  1. [Introduction] The notation for the generalized utility process U^c and the precise statement of the admissible consumption set should be introduced earlier and with explicit measurability requirements.
  2. [Examples] The examples section would benefit from a short table summarizing the hedging coefficients obtained under each return specification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Abstract] Abstract (regime statement): the entire variational problem, the definition of the objective functional via U^c, the perturbation argument, and the verification theorem presuppose that a unique generalized utility process exists for every non-negative progressively measurable consumption stream when θ∈(0,1). The manuscript invokes external uniqueness results for Epstein-Zin BSDEs but does not identify the precise theorem or set of conditions that guarantee uniqueness for arbitrary (including non-Markovian) streams; without this grounding the functional is not demonstrably well-defined on the whole admissible set.

    Authors: We agree that explicit identification of the uniqueness result is necessary for the functional to be well-defined on the full admissible set. The manuscript works throughout in the regime θ∈(0,1) and invokes known uniqueness results for Epstein-Zin BSDEs that apply to arbitrary progressively measurable consumption streams. We will revise the abstract and the opening paragraphs of the introduction to cite the precise theorem (including the conditions under which uniqueness holds for non-Markovian streams) so that the grounding is fully transparent. revision: yes

  2. Referee: [Verification theorem] Verification theorem (perturbation argument): the abstract states that the functional may be non-convex yet the direct method still yields existence, and that a perturbation argument identifies minimizers with the value function. No details are supplied on how the perturbation is constructed to avoid post-hoc choices or to obtain the comparison principle needed for optimality; this step is load-bearing for the claim that every minimizer solves the original control problem.

    Authors: The perturbation argument is indeed central to equating minimizers with the value function. It is constructed via a one-parameter family of admissible consumption processes under the myopic measure, combined with the uniqueness of the associated Epstein-Zin BSDE to obtain the required comparison. While the core construction appears in the verification section, we acknowledge that the exposition could be expanded to make the avoidance of post-hoc choices fully explicit. We will therefore provide a more detailed step-by-step description of the perturbation family and the resulting comparison principle in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external uniqueness results for Epstein-Zin BSDEs.

full rationale

The paper establishes a variational characterisation of the value function as the unique minimiser of a functional whose Euler-Lagrange equation coincides with the HJB equation, using a change of measure to the myopic probability together with uniqueness results for Epstein-Zin BSDEs and a perturbation argument. These uniqueness results are invoked as external mathematical facts rather than derived or fitted within the paper, and no load-bearing step reduces by construction to a self-defined quantity, a parameter fitted to the target result, or a self-citation chain whose authors overlap with the present work. The regime condition θ∈(0,1) for existence and uniqueness of the generalised utility process is stated as an assumption enabling the framework, not as a conclusion derived from the variational problem itself. The verification theorem and direct-method existence proof therefore remain independent of any internal redefinition or renaming of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the existence of a unique generalised utility process in the θ∈(0,1) regime and on cited uniqueness results for Epstein-Zin BSDEs; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence of a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams when θ∈(0,1)
    Invoked in abstract paragraph 2 to guarantee the utility is well-defined before the variational problem is posed.

pith-pipeline@v0.9.1-grok · 5733 in / 1403 out tokens · 22373 ms · 2026-06-28T11:17:27.968379+00:00 · methodology

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