Mathematical Reference
Ownership rule
Exact solutions and mathematical background belong in this section. A derivation, closed form, sign convention, process definition, or numerical method's mathematics is stated once here and referenced from elsewhere; it is never re-derived or restated. The solver, market_model, and engine pages document implementation only — contracts, configuration, and usage — and link back here for the mathematics they realize.
The single exception is the neural theory section, which carries its own method theory (deep BSDE, DGM, jump processes, neural operators) and is not folded into this section.
Canonical pages
The canonical mathematics lives in four self-contained pages, each stating its results once and without implementation detail:
- Stochastic Processes — the driving equations, parameters, and moments of every process (geometric Brownian motion, Ornstein-Uhlenbeck, CIR, Heston, jump diffusion, Bates, Hawkes, rough OU).
- Exact Solutions in Stochastic Optimal Control — the Avellaneda-Stoikov market-making family (CARA separation, reduced HJB, matrix-exponential solution, drift/impact and stationary limits), the Merton portfolio (diffusion, deterministic jumps, log-normal jumps), and the linear-quadratic regulator (correlated and with Poisson jumps).
- SOC Models Without Exact Solutions — SOC models without a closed form: the Heston stochastic-volatility and Hawkes order-flow reduced PDE/ODE systems and the American-put optimal-stopping problem, solved numerically and validated by degeneracy reduction.
- Exact Solutions to PDEs — the parabolic (heat, Black-Scholes, convection-diffusion) and elliptic (Laplace, Poisson, reaction-diffusion) benchmark problems used to validate the numerical solvers.
These pages carry the sign conventions and validation targets that the numerical solvers must reproduce. When a test, doc comment, or strategy needs a canonical formula, it links here rather than restating it.
Supplementary pages:
- Optimal Criteria — asymptotic approximations and the Kelly criterion, which are not exact solutions.
- Numerical Schemes — the finite-difference and BSDE methods whose convergence is validated against the exact solutions above.
- Backtest Framework — the simulation-layer accounting.
- Rigor Reference — the per-model mapping from problem to method and rigor level.
- Symmetries in Stochastic Optimal Control — a catalog of the symmetries and limit-induced symmetries of the SOC problems, cross-referencing the exact-solutions page.
- Limits of Stochastic Optimal Control Models — a catalog of the models obtained as limits of a more canonical model, cross-referencing the exact-solutions page.
- Cointegration — integrated and cointegrated series, the vector error-correction model and its Gaussian rank inference, the exact law of a mean-reverting relation, switching filters, the high-dimensional spectral laws, and the optimal stopping of a spread.
- Cointegration: Proof Obligations — the unproven statements the cointegration programme depends on, each with what must be proven, why, and what would refute it.
- Bibliography — citations.
The implementation-to-solution mapping (which code realizes which exact solution) is in the solver index.