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arxiv: 2506.22514 · v2 · submitted 2025-06-26 · 🧮 math.OC · math.PR

Optimal investment and consumption under forward utilities with relative performance concerns

Guillaume Broux-Quemerais (LMM) , Anis Matoussi (LMM) , Zhou Chao (NUS) This is my paper

Pith reviewed 2026-05-19 08:14 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords forward utilityrelative performanceCRRA utilityportfolio optimizationmean field gamesNash equilibriumstochastic HJB equationconsumption utility
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The pith

Forward relative performance processes with CRRA wealth utility necessarily have consumption utility of the same form and risk aversion.

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

The paper examines portfolio optimization problems for multiple players and in the mean-field limit where agents care about their performance relative to others. It focuses on forward utilities for both investment and consumption decisions. The key finding is that the consistency condition for these forward processes forces a specific link between the wealth utility and the consumption utility when the wealth part takes a constant relative risk aversion form with separable dependence on time and state. This link provides a clearer picture of what makes the strategies time-consistent over time. Readers should care because it offers explicit solutions for how competitive concerns shape both saving and spending in dynamic markets.

Core claim

The consistency assumption defining forward relative performance processes leads to a sufficient characterization of such processes with the mean of Stochastic HJB equations, which highlights the link between wealth and consumption utility. In particular, forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form, with the same risk aversion parameter. This characterization gives a better understanding of the drift condition ensuring time consistency. In this setting, closed forms of the Nash equilibrium are established for both the n-player and mean-field problems.

What carries the argument

The mean of Stochastic HJB equations arising from the consistency assumption on forward relative performance processes, which links the wealth and consumption utilities.

If this is right

  • Closed-form Nash equilibrium strategies for the finite n-player game with relative performance concerns.
  • Closed-form Nash equilibrium in the corresponding mean-field limit.
  • Explicit drift condition that ensures time consistency of the forward processes.
  • Numerical examples of optimal investment and consumption paths under the linked utilities.

Where Pith is reading between the lines

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

  • The same-form requirement may simplify solving dynamic consumption problems in other competitive market settings.
  • The mean-field limit supplies an approximation for large populations of investors facing relative performance concerns.
  • The utility link could be checked against data on whether consumption patterns align with wealth risk aversion in competitive environments.

Load-bearing premise

The consistency assumption defining forward relative performance processes is sufficient to characterize the processes via the mean of Stochastic HJB equations and to link wealth and consumption utilities.

What would settle it

A counter-example showing a forward relative performance process with CRRA wealth utility of separable time and space dependence but consumption utility of different form or risk aversion parameter would disprove the necessity of the link.

Figures

Figures reproduced from arXiv: 2506.22514 by Anis Matoussi (LMM), Guillaume Broux-Quemerais (LMM), Zhou Chao (NUS).

Figure 1
Figure 1. Figure 1: Kα,θ function of risk aversion α and competition parameter θ [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimal consumption for E log(K) = −0.5, E[θ] = 0.7, E [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimal consumption for K = 1.4 and |E log(K)| = 0.5 36 [PITH_FULL_IMAGE:figures/full_fig_p036_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimal consumption for K = 0.7 and |E log(K)| = 0.5 Relative market-consumption preference parameter - We now turn our attention to the case where utilities from wealth and consumption are non-linearly related through Assumption 3.4 as ϕt = Z 1−κ t . Taking κ < 0 leads to the logistic SDE with global solution, for which results on asymptotic behavior exist (see [20], [15]). In the following, we assume for… view at source ↗
Figure 7
Figure 7. Figure 7: Simulated paths of the equilibrium consumption rate process. Note the similarity between (4.6) and the strong MF equilibrium consumption, where this time the sign of the quantity R = b Z˜ − ∥δ Z ∥ 2 2 and its expectation reflects the agent and the population’s preference regarding consumption or wealth utility. In fact, • if R > 0 implies that log(Z˜) has positive drift so that Z˜ tends to increase in time… view at source ↗
read the original abstract

We study a n-player and mean-field portfolio optimization problem under relative performance concerns with non-zero volatility, for wealth and consumption. The consistency assumption defining forward relative performance processes leads to a sufficient characterization of such processes with mean of a Stochastic HJB equations, which highlights the link between wealth and consumption utility, and also characterizes the optimal strategies. In particular, forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form, with the same risk aversion parameter. This characterization gives a better understanding of the drift condition ensuring time consistency. In this setting, we establish closed form of the Nash equilibrium for both the n-player and mean eld problems. We also provide some numerical examples.

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 n-player and mean-field portfolio optimization problems involving both investment and consumption under relative performance concerns with non-zero volatility. Using a consistency assumption for forward relative performance processes, it provides a characterization through the mean of stochastic HJB equations that links the wealth and consumption utilities. In particular, it asserts that CRRA wealth utilities with separable time and space dependence imply consumption utilities of the same CRRA form with the same risk aversion parameter. Closed-form expressions for the Nash equilibria in both the finite and mean-field settings are derived, along with numerical examples.

Significance. If the central characterization holds, the paper offers a valuable contribution to stochastic control and mean-field games in finance by clarifying the structure of time-consistent forward utilities with relative concerns and delivering explicit optimal strategies. This could facilitate further analysis of competitive consumption-investment problems. The provision of closed-form solutions and numerical illustrations strengthens the practical relevance of the results.

major comments (2)
  1. [Abstract and characterization via mean of stochastic HJB] Abstract and the main characterization result: the claim that forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form with the same risk aversion parameter is asserted to follow from taking the mean of the stochastic HJB equations. However, with non-zero volatility in relative performance, the cross terms in the generator could permit other consumption utility forms (or different risk aversion) that still satisfy the drift condition for consistency, and the manuscript does not supply an explicit uniqueness argument for the resulting PDE system to exclude such possibilities.
  2. [Section on stochastic HJB mean and consistency condition] The link between wealth and consumption utilities is derived directly from the consistency assumption itself. To establish that this link is not tautological, an independent verification is required showing that the mean equation uniquely pins down the consumption utility to the claimed CRRA form whenever the wealth utility is separable CRRA, particularly when volatilities couple the processes non-separably.
minor comments (2)
  1. [Notation and preliminaries] The notation for the relative performance processes and the precise definition of the 'mean' operator applied to the stochastic HJB equations would benefit from additional explicit definitions early in the manuscript.
  2. [Numerical examples] In the numerical examples section, further discussion of how the equilibria respond to changes in the relative performance volatility parameters would improve interpretability of the figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the insightful comments on our paper. These have helped us clarify the presentation of our main results. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: Abstract and the main characterization result: the claim that forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form with the same risk aversion parameter is asserted to follow from taking the mean of the stochastic HJB equations. However, with non-zero volatility in relative performance, the cross terms in the generator could permit other consumption utility forms (or different risk aversion) that still satisfy the drift condition for consistency, and the manuscript does not supply an explicit uniqueness argument for the resulting PDE system to exclude such possibilities.

    Authors: We thank the referee for this observation. The mean of the stochastic HJB equations provides a deterministic PDE that the consumption utility must satisfy to ensure consistency. Given the CRRA form and separability in time and space for the wealth utility, this PDE can be solved explicitly, and the only solution that holds for the given structure, including the volatility cross terms, is the matching CRRA consumption utility with the same risk aversion. While the manuscript derives this directly, we acknowledge that an explicit uniqueness statement would strengthen the argument. We have revised the manuscript to include a brief uniqueness verification in the relevant section. revision: yes

  2. Referee: The link between wealth and consumption utilities is derived directly from the consistency assumption itself. To establish that this link is not tautological, an independent verification is required showing that the mean equation uniquely pins down the consumption utility to the claimed CRRA form whenever the wealth utility is separable CRRA, particularly when volatilities couple the processes non-separably.

    Authors: The consistency assumption is the starting point, but the specific form is pinned down by solving the resulting mean PDE under the separability hypothesis. The non-separable coupling through volatilities is incorporated into the generator, yet the CRRA structure ensures that the consumption utility must take the same form to balance the equation. To address the concern about tautology, we have added an independent verification step in the revised version, demonstrating uniqueness for the PDE system. revision: yes

Circularity Check

1 steps flagged

Consistency assumption directly yields consumption utility form via stochastic HJB mean

specific steps
  1. self definitional [Abstract]
    "The consistency assumption defining forward relative performance processes leads to a sufficient characterization of such processes with mean of a Stochastic HJB equations, which highlights the link between wealth and consumption utility... In particular, forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form, with the same risk aversion parameter."

    The consistency assumption is the definition of the forward processes; the paper then asserts that this assumption plus the mean of the stochastic HJB necessarily produces the identical CRRA consumption utility. The 'necessarily' claim therefore reduces to a restatement of the assumption's consequences rather than an independent derivation that could be falsified outside the assumption.

full rationale

The paper states that the consistency assumption defining forward relative performance processes leads to a characterization via the mean of stochastic HJB equations, which in turn forces the consumption utility to match the CRRA form and risk aversion of the wealth utility under separability. This link is presented as a necessary consequence, but the derivation reduces to the assumption itself enforcing the drift condition for time-consistency. No independent uniqueness theorem or external verification is quoted to show that other forms are excluded when relative performance volatility is non-zero. The central claim therefore has partial circular character as the result is largely by construction from the defining assumption rather than an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the consistency assumption for forward relative performance processes and on the existence of solutions to the associated Stochastic HJB equations; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Consistency assumption defining forward relative performance processes
    Invoked in the abstract as the starting point that leads to the Stochastic HJB characterization and the wealth-consumption link.

pith-pipeline@v0.9.0 · 5662 in / 1221 out tokens · 29481 ms · 2026-05-19T08:14:15.020033+00:00 · methodology

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Reference graph

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