Empirical Results
Measured results for the neural methods. The theory behind these methods is in the theory section.
Architecture-specific results
Each page focuses on finding the optimal network size/shape for each problem within a single architecture. Numbers are single-seed and indicative, not a rigorous hyperparameter optimum.
| Architecture | Page | Network |
|---|---|---|
| Deep BSDE | deep_bsde.md | \(Z\) (and \(\Gamma\)) nets |
| Deep HJB (DGM) | dgm.md | value net |
| Neural operator | operator.md | branch/trunk (DeepONet) or spectral (FNO) |
Cross-architecture results
| Page | Contents |
|---|---|
| Results | Validation, activation comparison, reference agreement |
| Comparison | Training/execution time and error vs exact, split PDE/SOC |
| Coverage | Which architecture fits each exact-solution problem |
Problems considered
| Problem | Class | Exact solution | Math background |
|---|---|---|---|
| Black-Scholes European call | PDE | closed form | BSDE/DGM theory; PDE exact (Black-Scholes) |
| Merton log-utility portfolio | SOC | closed form | SOC exact (Merton portfolio) |
| Linear-quadratic regulator | SOC | closed form (Riccati) | SOC exact (LQ regulator) |
| Jump linear-quadratic regulator | SOC (jump) | closed form (Riccati/affine) | SOC exact (LQ with jumps); jump theory |
| Avellaneda-Stoikov market making | SOC | closed form (matrix exponential) | SOC exact (Avellaneda-Stoikov) |
| Merton jump portfolio | SOC (jump) | closed form | SOC exact (Merton with deterministic jumps) |
| Merton jump log-normal | SOC (jump) | numerical root | SOC exact (Merton with log-normal jumps) |
The no-closed-form cases (Heston, Hawkes, correlated-default L3, unequal-\(k\)
market making) are not yet in neural_solver; they are the target of the
neural-operator stage
and are documented in the neural operator theory.