A Per-Component Diagnostic Protocol for Neural HJB-PIDE Solvers under Control-Dependent Lévy Jumps

Technical Summary: A Per-Component Diagnostic Protocol for Neural HJB-PIDE Solvers

Problem Statement
Neural solvers trained on residual losses are increasingly applied to Hamilton-Jacobi-Bellman partial integro-differential equations (HJB-PIDEs) arising in portfolio optimization under control-dependent Lévy jump dynamics. A persistent failure mode in this domain is that a learned solution may satisfy headline scalar diagnostics (e.g., matching the diffusion-only Merton ratio or minimizing the global loss) while systematically miscomputing the nonlocal integral operator within the training loss. Standard diagnostics—such as longer training, grid refinement, or truncation sweeps—often fail to detect these operator-level errors, particularly when the error manifests as a constant scaling factor in the importance proposal density used for Monte Carlo integration.

Methodology
The paper proposes a five-step per-component diagnostic protocol (Protocol 1) designed to isolate such failures by decomposing the Hamiltonian and comparing solvers against independent references before evaluating the final control policy. The protocol consists of:

1. Equation Standardization: Ensuring all compared solvers use the exact same HJB-PIDE, admissible set, and Lévy truncation bounds.
2. Independent Reference: Constructing at least one "from-scratch" reference implementation with disjoint numerical choices (e.g., different discretization, quadrature, or argmax routines).
3. Per-Component Decomposition: Tabulating the Hamiltonian's drift, diffusion, compensator, and nonlocal-integral terms separately across a grid of candidate controls ($u$). A constant ratio $\neq 1$ in any single component is identified as a signature of a proposal-density scale error.
4. Surface Comparison: Comparing the value function ($V$) and its low-order derivatives ($\partial_x V, \partial_{xx} V$) over a $(t, x)$ grid before comparing the argmax controls.
5. Delayed Trust: Only trusting the headline control match after the component and surface checks pass.

Experimental Setup and Bug Discovery
The protocol was applied to a standard Constant Relative Risk Aversion (CRRA) portfolio optimization problem with Variance Gamma (VG) jumps. While the CRRA homogeneity allows the problem to collapse to a scalar maximization (providing a semi-analytic baseline), the neural solver was tested against two independent finite-difference (FD) solvers and this scalar baseline.

Using the per-component diagnostic, the authors identified a critical implementation bug in their own neural solver:

• The Symptom: The neural solver's nonlocal integral and compensator terms were exactly half the values computed by the FD references across all candidate controls, while the drift and diffusion terms matched.
• The Root Cause: The importance sampling proposal density for the Lévy jumps was constructed as a 50/50 mixture of two one-sided shifted exponentials. The implementation calculated importance weights using the conditional density of a single tail ($q_{\pm}$) but omitted the $1/2$ mixture factor required for the unconditional density ($q = \frac{1}{2}q_{\pm}$). This resulted in the Monte Carlo estimator scaling the integral by exactly $1/2$.
• Invisibility: This error was invisible to standard diagnostics (longer training, grid refinement, loss curves) because the constant scaling factor did not prevent the network from converging to a local minimum that satisfied the residual loss relative to its own biased operator.

Key Results
After correcting the proposal density:

1. Convergence: Four independent references (two FD solvers, the corrected neural solver, and the semi-analytic scalar baseline) agreed on the optimal control ($u^*$) to within $\sim 2\%$ ($u^*_{VG} \approx 0.344$ vs. scalar baseline $0.352$).
2. Economic Sensitivity: The corrected solver demonstrated expected economic behaviors: a $\sim 54\%$ reduction in risky asset allocation compared to the diffusion-only Merton ratio due to negative-skew jump risk, and monotonic decreases in allocation with increasing jump intensity and negative skewness.
3. Surface Accuracy Limitations: Even after the fix, the neural solver's surface accuracy on the constant-coefficient benchmark was materially lower than the FD solvers. While the value function matched the reference to $0.6\%$, the second derivative ($\partial_{xx} V$) showed $\sim 26\%$ error, and the policy field varied by up to $15\%$ across the grid, despite the true optimal control being constant.

Significance and Claims
The paper makes a methodological contribution rather than a claim of a new high-dimensional solver or a convergence proof.

• The Protocol: The primary contribution is the diagnostic protocol itself, which is designed to be applicable to non-homogeneous and higher-dimensional settings where neural HJB-PIDE solvers are genuinely necessary (e.g., consumption with utility, transaction costs, regime-switching).
• Verification Failure Mode: The work highlights a broader failure mode in scientific machine learning: pointwise agreement of a learned value or control can coexist with a systematically wrong nonlocal operator.
• Modesty of Scope: The authors explicitly state that the CRRA-Merton-VG benchmark used is not a problem class where neural solvers are the most efficient method (due to homogeneity reduction). The benchmark serves solely as a test object to demonstrate the protocol's ability to catch a real, implementation-specific bug. The paper does not claim to have solved the general high-dimensional HJB theory, provided a viscosity-solution convergence proof, or demonstrated the protocol on non-homogeneous problems, though it posits this as the natural next step.

Reproducibility
The authors provide a full code release including the corrected neural solver, two FD reference solvers, the semi-analytic baseline, and the diagnostic scripts, allowing for the regeneration of all numeric claims.