Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations From Financial engineering to Active Matter

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The paper has moderate relevance to quantitative trading. It explicitly builds on the author's prior work on financial PIDEs (Lévy models, option pricing under stochastic volatility with correlated jumps). The backward Kolmogorov equation extension provides a discrete maximum principle for option prices. The positivity-preserving property is directly relevant for ensuring non-negative probability densities in risk models and valid option prices. However, the paper is primarily a numerical analysis contribution rather than a trading strategy or market microstructure paper. The financial applications are mentioned as motivation and extension rather than the primary focus.

High implementation complexity due to: (1) EM-matrix theory and spectral analysis requiring verification of Hypothesis H; (2) Strang splitting with multiple operator types (directional exponentials via Krylov, implicit cross-factor via factorized Picard); (3) Regime-switched stencils based on local Péclet number; (4) Krylov subspace methods with Arnoldi orthogonalization and scaling-and-squaring for small matrix exponentials; (5) Factorized Picard iteration with carefully chosen parameters P, Q, β; (6) Conservative flux closures for mass conservation; (7) Boundary condition handling (absorbing vs. reflecting); (8) Monitoring entrywise positivity at runtime. The 1D case is relatively straightforward, but the 2D scheme with cross-diffusion requires careful assembly of multiple interacting components.

The paper provides detailed mathematical derivations, complete algorithm descriptions (stencils, splitting, factorized solver, Krylov method), and mentions supporting Python code available on GitHub. Parameter settings for all numerical experiments are specified. However, the GitHub URL is not explicitly provided in the extract, and some proofs are deferred to appendices. The theoretical framework (EM-matrix theory, Hypothesis H) requires numerical verification for specific problems.