Classical Methods

This page records how the classical solver results are validated: the finite-difference (FD) and least-squares Monte Carlo (BSDE) paths in the Rust solver crate. For every benchmark problem it lists the reference used, the numerical method checked against it, and the tolerance at which the check currently passes. It complements the narrative solver pages (PDE Solving Methods and Elliptic PDE Solving Methods) by compressing the same information into machine-checkable tables. The neural methods are validated separately on the neural network methods page.

Validation policy

The solver has two kinds of validation problems:

Exact-solution problems have a closed-form or semi-analytical reference. They are validated numerical-versus-exact. Both the finite-difference (FD) and least-squares Monte Carlo (BSDE) paths are checked against the same reference where the method is applicable.
No-exact-solution problems (Hawkes, Heston, and higher-dimensional market-making models) have no reduced closed form. They are validated numerical-versus-numerical: FD against BSDE, plus degeneracy tests that reduce the model to a model with an exact solution.

Tolerances are recorded as obtained, not as goals. They reflect the assertions currently present in the tests. Every entry states whether the check is deterministic (FD, closed-form consistency) or stochastic (BSDE).

BSDE precision policy

BSDE is Monte Carlo and does not converge to high precision cheaply. Its standard-suite assertions therefore bound coarse, smoke-level error, not a tight tolerance. High-precision BSDE runs (targeting relative error at or below \(0.1\%\)) are a one-time exercise, not part of cargo test. They are run on demand and their results are recorded in the BSDE table below; they are not re-run on every build.

The standard BSDE checks divide into:

Tier 1 (default suite): fast smoke tests and zero-time-step consistency. These assert well-formedness, symmetry, and agreement at the terminal boundary, not tight finite-horizon accuracy.
Tier 2 (slow-tests): finite-horizon BSDE checks with bounded, generous tolerances driven by Monte Carlo noise.
Tier 3 (one-time exercise): high-precision BSDE runs targeting \(\leq 0.1\%\) relative error. Run explicitly, never in the default suite.

Reading the tolerance column

A percentage is a relative error bound.
An absolute figure is a spread or value error bound.
"Machine" means the check is at or near floating-point round-off.
"Finite/well-formed" means the assertion is qualitative (finite, positive, symmetric), not a numerical tolerance.

Canonical error metric

All numerical-versus-reference comparisons use the canonical error metric defined on the Validation page.

Exact-solution benchmarks: FD path

Problem	Reference	FD scheme	Obtained precision	Suite tier	Source
1D Laplace	\(u(x) = x\)	Stationary direct solve	\(10^{-6}\) rel	default	`elliptic_benchmarks.rs`
1D Poisson	\(u(x) = x(1-x)/2\)	Stationary direct solve	\(10^{-4}\) rel	default	`elliptic_benchmarks.rs`
2D separable Poisson	\(\sin(\pi x)\sin(\pi y)\)	Stationary SOR	\(2\%\) rel	default	`elliptic_benchmarks.rs`
1D reaction-diffusion	\(\sinh(\sqrt{a}\,x)\)	Stationary direct solve	\(10^{-4}\) rel	default	`elliptic_benchmarks.rs`
1D Neumann Laplace	\(u(x) = 1\)	Stationary direct solve	\(10^{-4}\) rel	default	`elliptic_benchmarks.rs`
1D Robin Laplace	\(u(x) = (x+1)/2\)	Stationary direct solve	\(10^{-4}\) rel	default	`elliptic_benchmarks.rs`
1D heat	\(\cos(\pi x)e^{-D\pi^2\tau}\)	Explicit / Implicit / CN	\(2\%\)-\(5\%\) rel	default	`pde_benchmarks.rs`
2D separable heat	\(\cos(\pi x)\cos(\pi y)e^{-2D\pi^2\tau}\)	Strang ADI vs fine explicit	\(1\%\) rel	default	`pde_schemes.rs`
1D convection-diffusion	traveling Gaussian	Implicit / CN	\(5\%\) rel	default	`pde_benchmarks.rs`
Black-Scholes call	Black-Scholes formula	Implicit / CN	\(5\%\) rel	default	`pde_benchmarks.rs`
American put (\(r=0\))	Black-Scholes put	Implicit / CN / Explicit	\(5\%\) rel	default	`american_put_fd.rs`, `pde_schemes.rs`
Merton portfolio	closed-form value and policy	Explicit	\(5\%\) rel	default	`merton_fd.rs`
Finite-horizon LQ	Riccati value and feedback	Explicit	\(10\%\) rel	default	`lq_regulator_fd.rs`
Correlated 2D LQ	Riccati value (full covariance)	Explicit	\(10\%\) rel	default	`lq_correlated_fd.rs`
Stationary LQ	algebraic Riccati	Policy iteration	\(10^{-3}\) rel	default	`elliptic_control.rs`
Avellaneda-Stoikov	matrix-exponential spreads	Explicit / Implicit / CN / ADI	\(0.05\) spread	default	`avellaneda_control_fd.rs`, `pde_schemes.rs`, `policy_iteration.rs`
Avellaneda drift	spectral exact spreads	Explicit	\(0.05\) spread	default	`avellaneda_drift_control_fd.rs`
Avellaneda impact	spectral exact spreads	Explicit	\(0.05\) spread	default	`avellaneda_impact_control_fd.rs`
Stationary Avellaneda	Gueant eigenvector and spreads	Perron power iteration	\(10^{-4}\)	default	`elliptic_avellaneda.rs`

Exact-solution benchmarks: BSDE path

BSDE is Monte Carlo. The standard assertions are smoke-level; the high-precision target is a one-time exercise whose obtained results are documented in BSDE Precision.

Problem	Reference	Standard tolerance	High-precision target	Suite tier	Source
Merton portfolio	closed-form policy/value	policy machine, value \(15\%\) rel	\(\leq 0.1\%\) rel	default	`merton_bsde.rs`
Finite-horizon LQ	Riccati policy/value	policy machine, value \(15\%\) rel	\(\leq 0.1\%\) rel	default	`lq_regulator_bsde.rs`
Avellaneda-Stoikov	matrix-exponential spreads	well-formed, symmetric	\(\leq 0.1\%\) rel	default	`avellaneda_control_bsde.rs`
Avellaneda-Stoikov	matrix-exponential spreads	\(10^{-4}\) total over \(q\)	\(\leq 0.1\%\) rel	`slow-tests`	`exact_vs_bsde/base.rs`
Avellaneda drift	spectral exact spreads	finite error	\(\leq 0.1\%\) rel	`slow-tests`	`exact_vs_bsde/drift.rs`
Avellaneda impact	spectral exact spreads	\(10^{-3}\) total over \(q\)	\(\leq 0.1\%\) rel	`slow-tests`	`exact_vs_bsde/impact.rs`

The BSDE high-precision target is not asserted in the default suite. The one-time high-precision verification and its obtained precision are recorded in BSDE Precision.

No-exact-solution benchmarks: numerical vs numerical

These models have no reduced closed form. FD and BSDE are checked against each other, and degeneracy limits reduce each model to an exact-solution reference. The expanded models (Heston, AvellanedaHawkes, HestonHawkes, BilateralHawkes, BilateralHawkesOrderFlowImbalance) are full-value problems: their bsde_driver is running_reward + local_source (factor transport excluded, reward rate bounded at 1/dt per fill side) and their next_step_controlled simulates inventory fills under the optimal control. Because their inventory dynamics depend on the control, finite-horizon BSDE checks run in coupled (Picard) mode; the decoupled mode is correct only at the terminal boundary.

Model	N	Cross-check	Obtained precision	Suite tier	Source
Heston	2	FD vs BSDE, zero time steps	\(10^{-5}\) spread	default	`bsde_vs_fdm/consistency.rs`
Heston	2	FD vs BSDE, finite horizon	\(0.05\) spread	`slow-tests`	`bsde_vs_fdm/consistency.rs`
Heston (\(v_{\xi}=0\))	2	FD vs AS exact, BSDE vs AS exact	FD \(10^{-3}\) spread, BSDE \(0.15\) spread	`slow-tests`	`bsde_heston/degenerate.rs`
Heston (\(v_{\xi}=0\), high \(a=140\))	2	BSDE vs AS exact, per-component bid/ask	\(0.02\) spread	`slow-tests`	`bsde_heston/exact_reduction.rs`
Hawkes	2	FD vs BSDE, zero time steps	\(10^{-5}\) spread	default	`bsde_vs_fdm/consistency.rs`
Hawkes	2	FD vs BSDE, finite horizon	\(0.15\) spread	`slow-tests`	`bsde_vs_fdm/consistency.rs`
Hawkes (\(\alpha=0\))	2	FD vs AS exact, BSDE vs AS exact	FD \(10^{-3}\) spread, BSDE \(0.10\) spread	`slow-tests`	`bsde_hawkes/degenerate.rs`
Hawkes (\(\alpha=0\), high \(\mu=140\))	2	BSDE vs AS exact, per-component bid/ask	\(0.02\) spread	`slow-tests`	`bsde_hawkes/exact_reduction.rs`
Bilateral Hawkes	3	FD vs BSDE, zero time steps	\(10^{-5}\) spread	default	`bsde_vs_fdm/bilateral_hawkes.rs`
Bilateral Hawkes	3	FD vs BSDE, finite horizon	\(5\%\) of base spread	`slow-tests`	`bsde_vs_fdm/bilateral_hawkes.rs`
Bilateral Hawkes (\(\alpha=0\), symmetric, high \(\mu=140\))	3	BSDE vs AS exact, per-component bid/ask	\(0.02\) spread	`slow-tests`	`bsde_bilateral_hawkes/exact_reduction.rs`
OFI (\(\eta_{\text{ofi}}=0\), \(\alpha=0\), symmetric, high \(\mu=140\))	3	BSDE vs AS exact, per-component bid/ask	\(0.02\) spread	`slow-tests`	`bsde_bilateral_hawkes/exact_reduction.rs`
Heston-Hawkes	3	degeneracy well-formedness	finite and positive	`slow-tests`	`bsde_heston_hawkes/degenerate.rs`
Heston-Hawkes (\(v_{\xi}=0\), \(\alpha=0\), high \(\mu=140\))	3	BSDE vs AS exact, per-component bid/ask	\(0.02\) spread	`slow-tests`	`bsde_heston_hawkes/exact_reduction.rs`
Manufactured reduced market-making	2	FD vs manufactured value, BSDE vs manufactured value	FD \(10^{-12}\) value, BSDE \(0.0025\) value	`slow-tests`	`manufactured_solution.rs`

Consistency and convergence checks

Beyond exact-vs-numerical and numerical-vs-numerical comparisons, the following checks pin down the analytical layer and solver convergence.

Check	Reference	Obtained precision	Suite tier	Source
Drift \(\mu=0\) reduces to base AS	`AvellanedaExact`	\(10^{-10}\) spread	default	`exact_vs_exact/consistency.rs`
Impact \(\xi=0\) reduces to base AS	`AvellanedaExact`	\(10^{-10}\) spread	default	`exact_vs_exact/consistency.rs`
Lot size \(=1\) reduces to base AS	`AvellanedaExact`	\(10^{-2}\) spread	default	`avellaneda_lot_size.rs`
Base FD time-refinement convergence	`AvellanedaExact`	monotone L2 decrease	default	`exact_vs_finite_diff/convergence_base.rs`
Drift FD time-refinement convergence	`AvellanedaDriftExact`	monotone L2 decrease	default	`exact_vs_finite_diff/convergence_drift.rs`
Impact FD time-refinement convergence	`AvellanedaImpactExact`	monotone L2 decrease	default	`exact_vs_finite_diff/convergence_market_impact.rs`
Policy iteration vs exact (implicit/explicit)	`AvellanedaExact`	\(0.1\%\)	default	`convergence/policy_iteration.rs`
Drift \(\mu=0\) vs exact	`AvellanedaExact`	\(2\%\)	default	`convergence/drift_zero.rs`
Impact \(\xi=0\) vs exact	`AvellanedaExact`	\(2\%\)	default	`convergence/impact_zero.rs`
American put convergence	Black-Scholes put	\(10\%\)	default	`convergence/american_put.rs`

Test suite tiers

Tier	Trigger	Contents	Approximate scope
Unit	`cargo test --lib -p solver`	Pure functions, no grids or solvers	always run
Default integration	`cargo test -p solver`	FD grid solves, analytical comparisons, smoke tests	always run
Slow	`cargo test -p solver --features slow-tests`	BSDE Monte Carlo, convergence sweeps, degeneracy	on demand
BSDE precision	manual, one-time	high-precision BSDE vs exact, \(\leq 0.1\%\)	never in CI

market-solve