{"id":"605c4303-b046-4d1f-bb04-7cd82207d9e4","arxiv_id":"2509.02267","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A PINN-based policy iteration scheme is proposed for a two-dimensional HJB equation combining proportional and stochastic-liquidity transaction costs, then tested on power, log, and exponential utilities.","lead":"The authors build a portfolio-choice model with two transaction-cost types, proportional fees and liquidity-driven market impact, and solve the resulting HJB equation with a deep-learning policy iteration algorithm. The paper reports that higher liquidity sensitivity and fee rates reduce risky holdings, and validates the solver in the zero-friction Merton limit.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence proof assumes (20), which is effectively the desired HJB limit; with only Merton-limit validation, the scheme is not shown to solve Eq. (12).","rationale":"The reader's weakest-assumption identification is accurate: Eq. (20) is the pivotal step in the convergence analysis and is effectively the desired conclusion. The proof shows the sequence is approximately monotone but does not establish that the policy-update step drives the sup-Hamiltonian residual to zero; this is assumed. Moreover, the only reported validation is the Merton limit (β=0, κ=0), which does not involve the two frictions that motivate the model. Therefore the paper's headline claim—that the proposed scheme solves the two-dimensional HJB equation with both cost types—is not backed by a valid proof or a full-model test. A finite-difference benchmark is the most direct way to test the claim: it bypasses the proof entirely and checks the output of the algorithm against a trusted numerical solution of Eq. (12). If the agreement is good, the central claim holds empirically and the proof's gap is a theoretical weakness; if the agreement is poor, the claim fails. I do not find an independent fatal flaw in the model derivation, although the fixed-δt treatment of transaction costs is heuristic; that is secondary to the missing verification. The reader's CONDITIONAL verdict is appropriate: the paper needs either a strengthened convergence argument or, at minimum, code and a full-model benchmark.","tokens_in":15599,"tokens_out":15829,"duration_ms":177067,"concrete_test":"Compute a reference solution of Eq. (12) by a high-order finite-difference or finite-element method on a bounded domain (W,L) ∈ [0.1,20]×[0.01,3], t∈[0,1], using the Table 1 parameters with a fixed ζ (e.g., ζ=0.5), and compare the converged PINN value function Q_φ and control ω_ψ against it. If the relative L2 error in Q is not below, say, 1%, then Algorithm 1 is not solving Eq. (12) in the advertised setting. This check settles the empirical validity of the central claim independently of the convergence proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3.3 aims to prove convergence of the deep policy iteration. The monotonicity argument (via Itô) gives V^{π_k} ≤ V^{π_{k+1}} + (T−t)||e_k||∞, but the limit being a solution of the HJB equation is secured only by the 'restrictive assumption' (20): H^{π_{k+1}}(t,x,V^{π_k}) → 0 uniformly. This quantity is the sup over controls of the Hamiltonian evaluated at the previous iterate; its vanishing is exactly the statement that the iterates satisfy the HJB equation in the limit. Thus the assumption is not a mild regularity condition but essentially the claimed convergence. No argument links the neural-network optimization (loss minimization over samples) to this uniform limit. The sole numerical validation (Section 3.4) takes β=0 and κ=0, reducing Eq. (12) to Merton's equation; it exercises neither the liquidity dimension nor the cost terms. Consequently, the central claim that Algorithm 1 solves Eq. (12) for the full two-dimensional problem rests on an unverified assumption plus a degenerate-case test.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15922,"tokens_out":4987,"duration_ms":60068,"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":[{"comment":"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.","section":"§3.3, Eq. (20)"},{"comment":"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.","section":"§2, Eqs. (5)–(9)"},{"comment":"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.","section":"§3.4 and §4"}],"minor_comments":[{"comment":"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).","section":"Algorithm 1, line 4"},{"comment":"The notation 'δWS' should be 'δB^S' (or 'dB^S') for consistency with the stock-price dynamics in Eq. (1).","section":"§2, Eq. (4)"},{"comment":"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.","section":"§3.4, Fig. 2"},{"comment":"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.","section":"§4.1, Eq. (22)"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The convergence analysis in §3.3 is the core theoretical contribution, but as written it assumes the conclusion. The numerical section validates only the frictionless limit. These issues are fixable in principle—by proving (20) in a meaningful setting or by adding a non-degenerate benchmark—but they are load-bearing, so I cannot recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my read. The genuinely new piece is the model in Section 2: an OU liquidity process whose long-run mean depends on the proportional cost rate, θ(L)=θ̄+κλL^ζ, plus the utility-maximization HJB with both frictions. That coupling is not in the papers they cite, and the resulting 2D HJB with a cross derivative is a legitimate target. The paper also gets credit for checking the zero-cost/zero-liquidity limit against Merton and for showing convergence of the iteration in that degenerate case. The neural-network policy iteration itself is not new — it is PINN plus Howard — but the application is sensible and the numerics look reasonably clean, with shaded bands over 10 runs.\n\nThe main problem is Section 3.3. The 'restrictive assumption' (20) says H^{π_{k+1}}(t,x,V^{π_k})→0 uniformly. That is not a regularity condition; it is the convergence claim. The argument up to that point only gives monotonicity up to an error term. So the theorem as stated does not prove convergence of the scheme; at best it shows convergence follows if the iterates solve the HJB in the limit. The paper cites Jacka–Mijatović, but no argument links the neural-network loss minimization to the uniform limit. This is a soft spot, and it is load-bearing because the abstract's 'convergence analysis in a general setting' is a headline claim.\n\nSecond, the only numerical validation sets β=0 and κ=0, which removes the liquidity dimension and the cost terms. It shows the implementation can recover Merton, but it does not test the model's new features. The full 2D runs are plots against Merton's line with no independent benchmark, no grid-based competitor, no known solution, and no code. The paper claims curse-of-dimensionality relief but shows only two state variables, and under CRRA the problem is actually reduced to 1D. ζ is never given a value, even though it appears in the model. There is also a small internal inconsistency: Eq. (15) sets the terminal condition at Q(W,L,T)=U(W), while Algorithm 1 writes Q(W,L,0)−U(W), and Eq. (9) has an extra δt factor.\n\nNone of this kills the underlying idea. The model is plausible and the numerical behavior in the figures matches economic intuition. But the advertised advantages — high-dimensional applicability and proven convergence — are not yet supported. What would change my mind: released code, a specified ζ, a nondegenerate validation (e.g., small transaction-cost asymptotics or a finite-difference benchmark in 2D), and either a real proof of (20) or an honest statement that convergence is conditional on it.\n\nBottom line: worth sending to a competent referee, but it needs substantive revision before publication. I would not cite it until the code and the nondegenerate test appear.","headline":"Novel model, plausible numerics, but convergence proof rests on an assumption that is the desired limit and validation only hits the Merton degenerate case.","tokens_in":16343,"tokens_out":2573,"would_cite":false,"duration_ms":27536,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91G10","49L25","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["portfolio selection","Hamilton-Jacobi-Bellman equation","liquidity risk","proportional transaction costs","policy iteration","deep learning","physics-informed neural networks","utility maximization"],"falsifier":"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.","tokens_in":15499,"feed_emoji":"📈","tokens_out":5431,"duration_ms":60453,"temperature":0.7,"pith_summary":"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.","feed_headline":"Deep learning scheme solves two-friction portfolio HJB equations","feed_subtitle":"Neural policy iteration handles proportional fees plus stochastic liquidity risk in one solver.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the mean-reverting liquidity model and correlated Brownian-motion setup that the paper adapts in its state dynamics.","marker":"[34]"},{"why":"Supplies the physics-informed neural network approach used to turn PDE residuals into a loss function.","marker":"[38]"},{"why":"Supplies the uniform-residual assumption under which the paper proves conditional convergence of the policy improvement step.","marker":"[24]"},{"why":"Supplies the frictionless Merton problem used as the analytical benchmark for numerical validation.","marker":"[32]"},{"why":"Supplies the classical policy iteration framework that the deep-learning scheme builds on.","marker":"[23]"},{"why":"Supplies the universal approximation result justifying the use of two-layer neural networks for the value and control functions.","marker":"[16]"},{"why":"Supplies the non-uniform collocation-point sampling idea used to weight residual regions more heavily in the loss.","marker":"[28]"},{"why":"Supplies the automatic differentiation technique used to compute derivatives without truncation error.","marker":"[5]"}],"fun_headline_variants":["Deep learning solves HJB with both cost types","Neural iteration handles high-dim portfolio HJB","Deep scheme removes truncation errors in HJB","AI method tackles two-cost portfolio optimization"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Deep learning solves HJB with both cost types","Neural iteration handles high-dim portfolio HJB","Deep scheme removes truncation errors in HJB","AI method tackles two-cost portfolio optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000414,"raw_usage":{"total_tokens":1930,"prompt_tokens":653,"completion_tokens":1277,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":397,"completion_tokens_details":{"reasoning_tokens":1220}},"tokens_in":397,"tokens_out":1277,"duration_ms":14086,"temperature":1.0,"reasoning_tokens":1220,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:42:36.452097+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Pasricha, S.-P","cited_arxiv_id":null,"evidence_quote":"Supplies the mean-reverting liquidity model and correlated Brownian-motion setup that the paper adapts in its state dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the uniform-residual assumption under which the paper proves conditional convergence of the policy improvement step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the frictionless Merton problem used as the analytical benchmark for numerical validation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical policy iteration framework that the deep-learning scheme builds on."},{"cited_title":"Funahashi","cited_arxiv_id":null,"evidence_quote":"Supplies the universal approximation result justifying the use of two-layer neural networks for the value and control functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the non-uniform collocation-point sampling idea used to weight residual regions more heavily in the loss."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the automatic differentiation technique used to compute derivatives without truncation error."}],"review_version":1}