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REVIEW 3 major objections 5 minor 39 references

A deep learning-driven iterative scheme for high-dimensional HJB equations in portfolio selection with exogenous and endogenous costs

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proposes a deep-learning-driven policy iteration scheme for the two-dimensional HJB equation arising when an investor faces both proportional transaction costs and stochastic liquidity risk, and shows numerically that the resultin

desk verdict Novel model, plausible numerics, but convergence proof rests on an assumption that is the desired limit and validation only hits the Merton degenerate case. read the letter →

arxiv 2509.02267 v1 pith:D2OGEVDA submitted 2025-09-02 q-fin.MF

Dong Yan , Nanyi Zhang , Junyi Guo This is my paper · ORCID
classification q-fin.MF MSC 91G1049L2568T07
keywords portfolioselectionHamilton-Jacobi-Bellmanequationliquidityriskproportionaltransactioncostspolicyiterationdeeplearningphysics-informedneuralnetworksutilitymaximization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies an investor who pays proportional transaction fees and also faces liquidity risk that moves stochastically, and wants to maximize expected utility of terminal wealth. The model leads to a nonlinear two-dimensional Hamilton-Jacobi-Bellman equation whose solution gives the optimal risky-asset weight. The paper claims that a policy iteration scheme in which both the value function and the control are represented by two-layer neural networks solves this equation without spatial grids or truncation error, and that the iteration is monotone and conditionally convergent. Numerical runs with power, logarithmic, and exponential utility show optimal risky weights below the no-friction benchmark, decreasing in the cost rate and in the sensitivity of asset prices to liquidity.

What carries the argument

The engine is a policy iteration loop over two neural networks. The value network Q_phi is trained so that the HJB operator applied under the current policy has small squared residual together with the terminal condition; the control network omega_psi, whose sigmoid output enforces the no-shorting constraint, is then trained to maximize that same operator against the updated value function. Derivatives come from automatic differentiation, so no finite-difference grid is needed, and convergence is argued provided the residual H^{π_{k+1}}(t,x,V^{π_k}) tends uniformly to zero, an assumption the paper adopts from earlier policy-improvement theory.

What would settle it

On the Merton validation case, compute the maximum over the sampled domain of |H^{π_{k+1}}(t,x,V^{π_k})| after each policy-improvement step: if this uniform residual does not decrease toward zero even when the value-function training loss shrinks, the central convergence hypothesis is not met. Alternatively, solve the full two-friction HJB with a dense grid-based finite-difference method on a fixed domain and compare the optimal-weight surface: disagreement beyond training noise would show the scheme solves a different limiting equation.

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Extended reading notes

Core claim

The central claim is that the deep learning-driven policy iteration scheme — alternating a policy-evaluation step that minimizes the squared PDE residual plus terminal-condition error for the value network, and a policy-improvement step that maximizes the HJB operator with respect to a sigmoid-constrained control network — produces accurate approximations of the value function and optimal policy for Eq. (12). The authors demonstrate this on the Merton benchmark and then use it to establish that both proportional fees and stochastic liquidity risk push the optimal stock fraction below the frictionless level, with the effect varying by utility type: for CRRA utilities the optimal weight is ind

Load-bearing premise

The convergence proof assumes that the residual H^{π_{k+1}}(t,x,V^{π_k}) goes to zero uniformly as iterations proceed, which is essentially the convergence the proof is trying to establish, and the paper gives no argument that the neural-network-trained policies satisfy this condition.

Editorial extensions

If this is right

  • The scheme is mesh-free, so adding another risky asset or another stochastic factor does not require re-gridding; the paper expects it to inherit the curse-of-dimensionality resistance of neural PDE solvers.
  • For CRRA utilities (power and log), the optimal risky weight is independent of time and wealth even with both frictions present, and the long-run mean-reversion level of liquidity does not affect the optimal weight.
  • For CARA (exponential) utility, the optimal weight is time-, wealth-, and liquidity-dependent; fees matter most for small wealth and become negligible for large wealth.
  • Raising either the proportional cost rate or the liquidity sensitivity lowers the optimal risky allocation below the frictionless benchmark.
  • Because the control is a neural network rather than a discretized set of grid values, the scheme extends naturally to problems with high-dimensional control spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same two-network iteration should apply to HJB equations with several risky assets or stochastic volatility, where the control network output dimension simply grows; the bottleneck will be whether policy improvement remains stable once the control space has many coordinates.
  • The paper's convergence theorem depends on the uniform residual condition (20) holding; a practical check would be to monitor that residual directly during training and use it as a stopping rule, turning the assumption into a testable diagnostic.
  • The finding that the long-run liquidity mean-reversion level does not affect CRRA allocations is stated for a power-function liquidity feedback; one could test whether it survives other concave choices of the feedback function.
  • The numerical comparisons are all against the frictionless Merton benchmark; comparing against a high-resolution finite-difference solution of the frictional HJB would test whether the PINN policy iteration also matches the true value function under frictions, not just in the benchmark limit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper considers a continuous-time portfolio selection problem with both exogenous proportional transaction costs and endogenous liquidity costs, the latter modeled by a mean-reverting Ornstein–Uhlenbeck process. The authors derive a two-dimensional HJB equation, propose a deep-learning-based policy iteration scheme (Algorithm 1), give a convergence argument in a general setting, and report numerical experiments for power, logarithmic, and exponential utility. The central claim is that the proposed scheme solves the full HJB equation (12), with potential advantages in high-dimensional settings and no truncation errors.

Significance. If the central claim were established, the paper would offer a practical mesh-free numerical method for high-dimensional HJB equations arising in portfolio selection with market frictions. The paper has clear strengths: the numerical scheme is tested against the Merton analytical solution in the degenerate case, the authors explicitly acknowledge the restrictive nature of their convergence assumption, and the three utility examples provide a useful qualitative picture of how liquidity risk and proportional costs affect portfolio choice. However, the main theoretical and numerical support for the full two-dimensional problem is currently incomplete, so the significance is conditional.

major comments (3)
  1. [§3.3, Eq. (20)] The convergence proof rests on the assumption H^{π_{k+1}}(t,x,V^{π_k}) → 0 uniformly. This is not a mild regularity condition; it is precisely the statement that the limit of the value iterates solves the HJB equation. The preceding monotonicity argument only shows that V^{π_k} is bounded below by the policy value and (under small errors) is increasing; it does not identify the limit as a solution of (18). No argument links the neural-network loss minimization in Algorithm 1 to this uniform residual limit. Thus the convergence theorem, as stated, is essentially circular and does not establish that Algorithm 1 solves Eq. (12). I would ask the authors to either prove (20) under verifiable conditions on the control and value networks, or invoke and verify a known policy-iteration convergence result, and to report the HJB residual in the numerical experiments.
  2. [§2, Eqs. (5)–(9)] The derivation of the traded quantity ν and the expected transaction cost is heuristic and contains a dimensionally suspicious expression. Combining Eq. (6) with Eq. (8) gives an expected cost proportional to √δt, not δt, because Eq. (8) already contains √δt. Yet Eq. (9) and the wealth dynamics (10) are written with an extra δt factor, i.e. the cost term is proportional to √(2/(πδt))·δt. Since the cost term enters the drift of the HJB equation, this discrepancy changes the model and the numerical results. Please clarify the approximation and correct the formula. Also, Eq. (5) introduces κ·sign(ν) in the denominator without derivation; the subsequent approximation in Eq. (7) drops the sign dependence, which should be justified rather than assumed.
  3. [§3.4 and §4] The only quantitative validation of the scheme is the Merton case with β=0 and κ=0, where the two-dimensional HJB equation (12) reduces to a one-dimensional problem without cost terms. This test does not exercise the liquidity dimension, the stochastic-liquidity dynamics, or the transaction-cost terms. The full two-dimensional results in Section 4 are presented without comparison to any reference solution. I recommend adding a validation against a standard finite-difference or alternative numerical solver for at least one of the examples with nonzero β and κ; for Example 1, the power-utility reduction to one dimension provides a natural benchmark.
minor comments (5)
  1. [Algorithm 1, line 4] The terminal condition is written as |Qϕ(W,L,0)-U(W)|², but the problem is defined on [0,T]; the correct condition is at t=T, as in Eq. (15).
  2. [§2, Eq. (4)] The notation 'δWS' should be 'δB^S' (or 'dB^S') for consistency with the stock-price dynamics in Eq. (1).
  3. [§3.4, Fig. 2] The claim of 'exponential or even faster convergence' is based on an empirical log-distance plot for the Merton case; this should be described as empirical, not as a proven rate, and ideally accompanied by a quantified rate or error bars.
  4. [§4.1, Eq. (22)] The reduction to a one-dimensional PDE for power utility is useful, but the paper does not use this reduced PDE as a benchmark for the general 2D solver. This is a missed opportunity and should be discussed.
  5. [Throughout] The parameter δt is fixed at 1/12 without a sensitivity study. Since the model depends on δt through the transaction-cost approximation, a robustness check over δt would strengthen the numerical evidence.

Circularity Check

1 steps flagged · score 6.0 of 10

Convergence proof assumes (20), which is the HJB limit itself; the central theoretical claim is conditional on its own conclusion.

  1. other [Section 3.3, Convergence Analysis, assumption (20)]
    "Moreover, the convergence of our numerical scheme can be proved under the following restrictive assumption: H πk+1(t, x, Vπk) → 0 uniformly, (20), which was first proposed by Jacka and Mijatovi´ c [24]."

    By Eq. (19), H^{π}(t,x,v)=∂_t v+H(t,x,∇v,∇²v,π(t,x)); and the policy-improvement step gives H^{π_{k+1}}(t,x,V^{π_k})=sup_{u∈U}H^u(t,x,V^{π_k}). Thus (20) says exactly that the approximate value functions V^{π_k} satisfy the HJB equation (18) in the uniform limit—the very convergence the section claims to prove. The preceding Itô argument only yields V^{π_k}≤V^{π_{k+1}}+(T−t)||e_k||∞ and does not derive (20) from Algorithm 1. The 'restrictive assumption' is therefore not a mild regularity condition but the target HJB limit itself, so the convergence theorem assumes rather than establishes the desired result.

full rationale

The paper's numerical scheme has independent content: Section 3.4 validates the solver against Merton's analytical solution, and the examples are forward simulations with fixed parameters rather than fits to the target solution. However, the central theoretical claim — that Algorithm 1 converges to the solution of the two-dimensional HJB equation (12) — rests on assumption (20), which is effectively the HJB residual vanishing in the limit. The paper does not show that the neural-network loss minimization produces this uniform limit; it merely labels the condition 'restrictive' and cites [24]. In addition, the only benchmark used (Merton with β=0 and κ=0) is a degenerate case that does not exercise the liquidity or transaction-cost terms. The full-model claim therefore depends on a circular/conditional convergence proof. No self-citation chain, ansatz-smuggling, or renaming of known results is present; the circularity is concentrated in the convergence analysis.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the HJB model, the neural-network approximation, and a convergence assumption that is effectively the limit being proved. The two new coupling parameters λ and ζ are uncalibrated, and ζ is never specified in the numerical experiments.

free parameters (2)
  • λ (transaction cost sensitivity in θ(L)) = 5.0
    Chosen by hand in Table 1 to set θ(L)=θ̄+κλL^ζ; no calibration or sensitivity analysis is provided.
  • ζ (exponent in g(L)=L^ζ) = unspecified, claimed in (0,1)
    Defines the 'intrinsic relationship' between exogenous and endogenous costs, but its numerical value is not given anywhere in the paper, so the experiments are not fully reproducible from the text.
assumptions (6)
  • domain assumption The HJB equation (12) admits a classical solution and two-layer neural networks can approximate Q* and ω* sufficiently well.
    Invoked in Sections 3.1 and 3.2; no regularity or approximation error proof is given for this specific HJB.
  • domain assumption κ^2ω << 1, so κ/(1+κ sign(ν)ω) ≈ κ.
    Used in Eq. (7) to simplify expected transaction costs; numerically true for the small κ used, but it is an uncontrolled approximation that changes the HJB generator.
  • domain assumption V^{π_k} and f^{π_k} grow quadratically in x and the localizing argument justifies dominated convergence.
    Stated in Section 3.3; standard in control proofs but unverified for the trained neural-network functions.
  • ad hoc to paper The restrictive convergence assumption H^{π_{k+1}}(t,x,V^{π_k}) → 0 uniformly holds.
    Eq. (20) is effectively the convergence limit being proved; the paper provides no argument that the trained policies satisfy it.
  • domain assumption The stock-price and liquidity dynamics (1) from Pasricha et al. are taken as given.
    The model is borrowed from prior work [34] without empirical validation in this paper.
  • ad hoc to paper The coupling θ(L)=θ̄+κλL^ζ represents the intrinsic relationship between exogenous and endogenous transaction costs.
    Introduced in Eq. (3) without empirical support; ζ is unspecified and λ is arbitrary.

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Pith. "Pith review of A deep learning-driven iterative scheme for high-dimensional HJB equations in portfolio selection with exogenous and endogenous costs." pith.science (2026). https://pith.science/paper/D2OGEVDA

@misc{pith2026250902267,
  author       = {Pith},
  title        = {Pith review of: A deep learning-driven iterative scheme for high-dimensional HJB equations in portfolio selection with exogenous and endogenous costs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D2OGEVDA}},
  note         = {Machine review of arXiv:2509.02267}
}
read the original abstract

In this paper, we first conduct a study of the portfolio selection problem, incorporating both exogenous (proportional) and endogenous (resulting from liquidity risk, characterized by a stochastic process) transaction costs through the utility-based approach. We also consider the intrinsic relationship between these two types of costs. To address the associated nonlinear two-dimensional Hamilton-Jacobi-Bellman (HJB) equation, we propose an innovative deep learning-driven policy iteration scheme with three key advantages: i) it has the potential to address the curse of dimensionality; ii) it is adaptable to problems involving high-dimensional control spaces; iii) it eliminates truncation errors. The numerical analysis of the proposed scheme, including convergence analysis in a general setting, is also discussed. To illustrate the impact of these two types of transaction costs on portfolio choice, we conduct through numerical experiments using three typical utility functions.

Figures

Figures reproduced from arXiv: 2509.02267 by the authors.

Figure 1
Figure 1. Validation of numerical scheme with zero liquidity risk and zero transaction costs [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Experimental convergence rates with zero liquidity risk and zero transaction costs [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. The variation with time for W = 2.5 and L = 0.6. 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ β = 0.1 β = 0.3 β = 0.5 Merton’s line (a) Different β 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ κ = 0.0% κ = 0.4% κ = 0.8% Merton’s line (b) Different κ [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: The variation with wealth for L = 0.6 and t = 0.5. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: The variation with liquidity risk for W = 2.5 and t = 0.5. 0.0 0.2 0.4 0.6 0.8 1.0 Time t (years) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ θ¯ = 0.2 θ¯ = 0.6 θ¯ = 1.0 Merton’s line 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8…
Figure 6
Figure 6. Figure 6: The variation with ¯θ for W = 2.5, L = 0.6 and t = 0.5. Note that the shaded region shown in the figures above represents the results across 10 independent runs, demonstrating the stability and robustness of our algorithm. When either exogenous or endogenous transactio…
Figure 7
Figure 7. Figure 7: The variation with time for W = 2.5 and L = 0.6. 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ β = 0.1 β = 0.3 β = 0.5 Merton’s line (a) Different β 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal…
Figure 8
Figure 8. Figure 8: The variation with wealth for L = 0.6 and t = 0.5. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: The variation with liquidity risk for W = 2.5 and t = 0.5. 0.0 0.2 0.4 0.6 0.8 1.0 Time t (years) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ θ¯ = 0.2 θ¯ = 0.6 θ¯ = 1.0 Merton’s line 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8…
Figure 10
Figure 10. Figure 10: The variation with ¯θ for W = 2.5, L = 0.6 and t = 0.5. In this case, the results are similar to those of Example 1. A larger value of β, reflecting an increased effect of liquidity risk on the underlying asset, correlates with a reduced proportion of the portfolio in…
Figure 11
Figure 11. Figure 11: The variation with time for W = 2.5 and L = 0.6. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: The variation with wealth for L = 0.6 and t = 0.5. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Liquidity Risk L 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ β = 0.1 β = 0.3 β = 0.5 Merton’s line (a) Different β 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0…
Figure 13
Figure 13. Figure 13: The variation with liquidity risk for W = 2.5 and t = 0.5. 0.0 0.2 0.4 0.6 0.8 1.0 Time t (years) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Optimal investement policy ω ∗ θ¯ = 0.2 θ¯ = 0.6 θ¯ = 1.0 Merton’s line 2 3 4 5 6 7 8 9 10 Wealth W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.…
Figure 14
Figure 14. Figure 14: The variation with ¯θ for W = 2.5, L = 0.6 and t = 0.5. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

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