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arxiv: 2502.11052 · v2 · submitted 2025-02-16 · 🧮 math.OC · q-fin.PM

Time-consistent portfolio selection with monotone mean-variance preferences

Yike Wang , Yusha Chen , Jingzhen Liu This is my paper

Pith reviewed 2026-05-23 03:04 UTC · model grok-4.3

classification 🧮 math.OC q-fin.PM
keywords time-inconsistent portfolio selectionmonotone mean-variance preferencesNash equilibriumopen-loop controlclosed-loop controlforward-backward stochastic differential equations
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The pith

Monotone mean-variance portfolio selection yields time-consistent Nash equilibria with higher risky asset holdings than mean-variance selection.

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

The paper investigates portfolio selection problems that become time-inconsistent when using monotone mean-variance preferences because their optima match those of standard mean-variance preferences. It resolves the inconsistency by deriving both open-loop and closed-loop Nash equilibrium strategies through systems of forward-backward stochastic differential equations and extended Hamilton-Jacobi-Bellman equations. Under deterministic parameters these equilibria admit the same semi-closed-form expression that does not depend on current wealth or the realized market path. The amount invested in the risky asset is larger than the corresponding mean-variance amount, with the difference decreasing as time progresses. For constant parameters the open-loop equilibrium is always a strong equilibrium while the closed-loop one is strong only if the interest rate is sufficiently high.

Core claim

The central discovery is that the time-inconsistency arising from monotone mean-variance preferences in portfolio choice can be addressed by open-loop and closed-loop Nash equilibria whose semi-closed-form solutions under deterministic parameters are independent of wealth and take the same form, resulting in greater investment in risky assets than mean-variance strategies, and with the open-loop version always qualifying as a strong equilibrium under constant parameters.

What carries the argument

Open-loop and closed-loop Nash equilibrium controls characterized by flows of forward-backward stochastic differential equations and systems of extended Hamilton-Jacobi-Bellman equations.

If this is right

  • The investment in the risky asset under the MMV equilibrium exceeds that under the MV equilibrium.
  • The gap between MMV and MV investment amounts narrows over time.
  • Under constant parameters the open-loop Nash equilibrium control is a strong equilibrium strategy.
  • The closed-loop Nash equilibrium control is a strong equilibrium strategy only when the interest rate is sufficiently large.

Where Pith is reading between the lines

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

  • If the coincidence of MMV and MV optima holds more generally, similar Nash resolutions could apply to other preference classes.
  • The wealth-independence of the strategy suggests that equilibrium investment proportions remain constant regardless of current wealth levels.
  • The narrowing investment gap implies that MMV and MV strategies converge as the horizon shortens.

Load-bearing premise

Optimal strategies for monotone mean-variance preferences coincide with those for mean-variance preferences.

What would settle it

A direct computation or simulation demonstrating that the derived closed-loop control fails to satisfy the strong equilibrium condition for a small interest rate value.

Figures

Figures reproduced from arXiv: 2502.11052 by Jingzhen Liu, Yike Wang, Yusha Chen.

Figure 1
Figure 1. Figure 1: Equilibrium investment amount in the risky asset with different preferences. The parameters [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Equilibrium investment amount in the risky asset under the SMMV preference w.r.t. different [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

We investigate time-inconsistent portfolio problems under a broader class of monotone mean-variance (MMV) preferences. Since the optimal strategies for MMV and mean-variance (MV) preferences coincide, the MMV optimal strategies at different initial times are necessarily time-inconsistent. To address this time-inconsistency, we consider Nash equilibrium controls of both open-loop and closed-loop types, and characterize them within a random parameter setting. The two control problems reduce to solving a flow of forward-backward stochastic differential equations and a system of extended Hamilton-Jacobi-Bellman equations, respectively. In particular, we derive semi-closed-form solutions for both types of equilibria under a deterministic parameter setting, and both solutions share the same representation, which is independent of the wealth state and the random path. We show that the investment amount under the MMV equilibrium exceeds that under the MV equilibrium, and the gap narrows over time. Furthermore, under a constant parameter setting, we find that the derived closed-loop Nash equilibrium control is a strong equilibrium strategy only when the interest rate is sufficiently large, whereas the derived open-loop Nash equilibrium control is necessarily a strong equilibrium strategy.

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 / 1 minor

Summary. The manuscript investigates time-inconsistent portfolio selection under monotone mean-variance (MMV) preferences. It asserts that optimal strategies coincide with those under mean-variance (MV) preferences, leading to time-inconsistency addressed via open- and closed-loop Nash equilibria. These are characterized using flows of FBSDEs and extended HJBs. Semi-closed-form solutions are derived under deterministic parameters, independent of wealth and paths, showing higher investment under MMV than MV with narrowing gap. Under constant parameters, closed-loop equilibria are strong only for sufficiently large interest rates, while open-loop are always strong.

Significance. If the central claims hold, the work extends time-consistent portfolio theory to MMV preferences with explicit equilibrium characterizations that are wealth- and path-independent. The semi-closed-form solutions and the distinction between open- and closed-loop strong-equilibrium conditions under constant parameters constitute a technical contribution to dynamic optimization in finance.

major comments (2)
  1. [Abstract] Abstract: The premise that 'the optimal strategies for MMV and MV preferences coincide' is asserted without derivation or citation. This is load-bearing for framing the problem as time-inconsistent and for the subsequent reduction to Nash equilibria via FBSDEs and extended HJBs; verification that monotonicity does not alter the first-order conditions is required.
  2. [Constant parameter setting] Constant parameter setting (results on strong equilibria): The statement that the closed-loop Nash equilibrium control is a strong equilibrium only when the interest rate is sufficiently large lacks an explicit threshold or derivation tying the bound to the FBSDE solution; the open-loop claim of being necessarily strong also requires the supporting inequality or condition to be shown explicitly.
minor comments (1)
  1. [Abstract] The abstract mentions a 'random parameter setting' for the general characterization but does not clarify how the deterministic case specializes; a brief sentence on the reduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The premise that 'the optimal strategies for MMV and MV preferences coincide' is asserted without derivation or citation. This is load-bearing for framing the problem as time-inconsistent and for the subsequent reduction to Nash equilibria via FBSDEs and extended HJBs; verification that monotonicity does not alter the first-order conditions is required.

    Authors: We agree that an explicit verification is required. The coincidence follows because MMV preferences are obtained via a strictly increasing transformation of the MV objective, which leaves the argmax unchanged and thus preserves the first-order conditions. In the revised manuscript we will insert a short lemma (or paragraph in Section 2) deriving this fact directly from the definition of the MMV functional. revision: yes

  2. Referee: [Constant parameter setting] Constant parameter setting (results on strong equilibria): The statement that the closed-loop Nash equilibrium control is a strong equilibrium only when the interest rate is sufficiently large lacks an explicit threshold or derivation tying the bound to the FBSDE solution; the open-loop claim of being necessarily strong also requires the supporting inequality or condition to be shown explicitly.

    Authors: We accept the criticism. In the revision we will derive the explicit lower bound on the interest rate from the FBSDE coefficients and state the threshold condition clearly for the closed-loop case. For the open-loop case we will supply the explicit inequality (derived from the FBSDE solution) that holds for all admissible parameter values, thereby confirming that the equilibrium is always strong. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on standard stochastic control reductions from stated premise

full rationale

The paper takes as premise that MMV and MV optima coincide (abstract), which induces time-inconsistency and motivates the Nash-equilibrium formulation. It then reduces both open- and closed-loop problems to a flow of FBSDEs and a system of extended HJBs, respectively, and obtains semi-closed-form solutions under deterministic parameters that are independent of wealth and paths. These steps use conventional forward-backward techniques without any quoted self-definitional equivalence, fitted parameter renamed as prediction, or load-bearing self-citation chain. The subsequent claims about investment amounts, gaps narrowing over time, and strong-equilibrium conditions (interest-rate threshold for closed-loop) are derived from the equilibrium equations rather than presupposed by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no identifiable free parameters, axioms, or invented entities; model relies on standard stochastic control setup but specifics unavailable.

pith-pipeline@v0.9.0 · 5731 in / 1116 out tokens · 38001 ms · 2026-05-23T03:04:24.736260+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Basak, S., & Chabakauri, G. (2010). Dynamic mean-variance asset allocation.The Review of Financial Studies, 23(8), 2970-3016. doi: 10.1093/rfs/hhq028 Björk, T., Khapko, M., & Murgoci, A. (2017). On time-inconsistent stochastic control in continuous time. Finance and Stochastics, 21(2), 331-360. doi: 10.1007/s00780-017-0327-5 Björk, T., Murgoci, A., & Zhou...

    work page doi:10.1093/rfs/hhq028 2010

  2. [2]

    J., & Xia, Y

    Brennan, M. J., & Xia, Y. (2002). Dynamic asset allocation under inflation.The Journal of Finance, 57(3), 1201-1238. doi: 10.1111/1540-6261.00459

    work page doi:10.1111/1540-6261.00459 2002

  3. [3]

    Goldman, S. M. (1980). Consistent plans.Review of Economic Studies, 47(3), 533-537. doi: 10.2307/ 2297304

    work page 1980

  4. [4]

    He, X., & Jiang, Z. (2021). On the equilibrium strategies for time-inconsistent problems in continuous time. SIAM Journal on Control and Optimization, 59(5), 3860-3886. doi: 10.1137/20M1382106

    work page doi:10.1137/20m1382106 2021

  5. [5]

    Hu, Y., Jin, H., & Zhou, X. (2012). Time-inconsistent stochastic linear-quadratic control.SIAM Journal on Control and Optimization, 50(3), 1548-1572. doi: 10.1137/110853960

    work page doi:10.1137/110853960 2012

  6. [6]

    Hu, Y., Jin, H., & Zhou, X. (2017). Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium.SIAM Journal on Control and Optimization, 52(2), 1261-1279. doi: 10.1137/15M1019040

    work page doi:10.1137/15m1019040 2017

  7. [7]

    Hu, Y., Shi, X., & Xu, Z. (2023). Constrained monotone mean-variance problem with random coefficients. SIAM Journal on Financial Mathematics, 14(3), 838-854. doi: 10.1137/22m154418x

    work page doi:10.1137/22m154418x 2023

  8. [8]

    (2009).Mathematical methods for financial market

    Jeanblanc, M., Yor, M., & Shesney, M. (2009).Mathematical methods for financial market. London: Springer-Verlag. doi: 10.1007/978-1-84628-737-4

    work page doi:10.1007/978-1-84628-737-4 2009

  9. [9]

    Li, B., & Guo, J. (2021). Optimal reinsurance and investment strategies for an insurer under monotone mean-variance criterion.Rairo-Operations Research, 55(4), 2469-2489. doi: 10.1051/ro/2021114

    work page doi:10.1051/ro/2021114 2021

  10. [10]

    Li, B., Guo, J., & Tian, L. (2023). Optimal investment and reinsurance policies for the cramér-lundberg risk model under monotone mean-variance preference.International Journal of Control, 97(6), 1296-1310. doi: 10.1080/00207179.2023.2204384

    work page doi:10.1080/00207179.2023.2204384 2023

  11. [11]

    Li, D., & Ng, W. (2000). Optimal dynamic portfolio selection: Multi-period mean-variance formulation. Mathematical Finance, 10(3), 387-406. doi: 10.1111/1467-9965.00100

    work page doi:10.1111/1467-9965.00100 2000

  12. [12]

    Li, Y., Liang, Z., & Pang, S. (2022). Continuous-time monotone mean-variance portfolio selection. arXiv:2211.12168v5. doi: 10.48550/arXiv.2211.12168

    work page doi:10.48550/arxiv.2211.12168 2022

  13. [13]

    Li, Y., Liang, Z., & Pang, S. (2023). Comparison between mean-variance and monotone mean-variance preferences under jump diffusion and stochastic factor model.arXiv:2211.14473v2. doi: 10.48550/ arXiv.2211.14473

    work page arXiv 2023

  14. [14]

    Liang, Z., Xia, J., & Yuan, F. (2023). Dynamic portfolio selection for nonlinear law-dependent preferences. arXiv:2311.06745v2, 1-36. doi: 10.48550/arXiv.2311.06745

    work page doi:10.48550/arxiv.2311.06745 2023

  15. [15]

    Maccheroni, F., Marinacci, M., & Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences.Econometrica, 74(6), 1447-1498. doi: 10.1111/J.1468-0262 .2006.00716.X

    work page doi:10.1111/j.1468-0262 2006

  16. [16]

    Maccheroni, F., Marinacci, M., Rustichini, A., & Taboga, M. (2009). Portfolio selection with monotone mean-variance preferences.Mathematical Finance, 19(3), 487-521. doi: 10.1111/j.1467-9965.2009 .00376.x

    work page doi:10.1111/j.1467-9965.2009 2009

  17. [17]

    Markowitz, H. (1952). Portfolio selection.Journal of Finance, 7(1), 77-91. doi: 10.2307/2975974

    work page doi:10.2307/2975974 1952

  18. [18]

    Peleg, B., & Yaari, M. E. (1973). On the existence of a consistent course of action when tastes are changing. Review of Economic Studies, 40(3), 391-401. doi: 10.2307/2296458 24

    work page doi:10.2307/2296458 1973

  19. [19]

    Pollak, R. A. (1968). Consistent planning.Review of Economic Studies, 35(2), 201-208. doi: 10.2307/ 2296548

    work page 1968

  20. [20]

    Shen, Y., & Zou, B. (2022). Cone-constrained monotone mean-variance portfolio selection under diffusion models. arXiv:2205.15905. doi: 10.48550/arXiv.2205.15905

    work page doi:10.48550/arxiv.2205.15905 2022

  21. [21]

    Strotz, R. H. (1955). Myopia and inconsistency in dynamic utility maximization.Review of Economic Studies, 23(3), 165-180. doi: 10.1007/978-1-349-15492-0_10

    work page doi:10.1007/978-1-349-15492-0_10 1955

  22. [22]

    S., & Li, D

    Strub, M. S., & Li, D. (2020). A note on monotone mean-variance preferences for continuous processes. Operations Research Letters, 48(4), 397-400. doi: 10.1016/j.orl.2020.05.003

    work page doi:10.1016/j.orl.2020.05.003 2020

  23. [23]

    Trybula, J., & Zawisza, D. (2019). Continuous-time portfolio choice under monotone mean-variance preferences–stochastic factor case.Mathematics of Operations Research, 44(3), 966-987. doi: 10.1287/MOOR.2018.0952 ˘Cerný, A. (2020). Semimartingale theory of monotone mean-variance portfolio allocation.Mathematical Finance, 30(3), 1168-1178. doi: 10.1111/mafi.12241

    work page doi:10.1287/moor.2018.0952 2019

  24. [24]

    Wang, Y., & Chen, Y. (2024). Strictly monotone mean-variance preferences with dynamic portfolio management. arXiv:2412.13523, 1-52. doi: 10.48550/arXiv.2412.13523

    work page doi:10.48550/arxiv.2412.13523 2024

  25. [25]

    Wang, Y., Liu, J., Bensoussan, A., Yiu, K. F. C., & Wei, J. (2025). On stochastic control problems with higher-order moments.SIAM Journal on Control and Optimization, in press. doi: 10.48550/ arXiv.2412.13521

    work page arXiv 2025

  26. [26]

    Yong, J. (2012). Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control and Related Fields, 2(3), 271-329. doi: 10.3934/mcrf.2012.2.271

    work page doi:10.3934/mcrf.2012.2.271 2012

  27. [27]

    Zhou, X., & Li, D. (2000). Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Applied Mathematics and Optimization, 42(1), 19-33. doi: 10.1007/s002450010003 25

    work page doi:10.1007/s002450010003 2000