Diagonal Frog meets ADI: trading matrix exponentials for rational maps in the Fokker–Planck equation

Battle Count: 54

Highly relevant to quantitative trading and financial engineering. The paper directly addresses the Fokker-Planck equation for anisotropic diffusion (cross-diffusion), which is central to multi-asset option pricing under Heston-type models with stochastic volatility and correlation. The jump-diffusion extension covers Merton and Kou models used in practice. The backward Kolmogorov equation extension is directly applicable to derivative pricing. The positivity preservation is critical for probability density functions in risk management. The O(N) cost per step makes it practical for real-time pricing. However, it is a numerical method paper rather than a trading strategy paper, so relevance is at the infrastructure/solver level rather than the strategy level.

The scheme requires: (1) assembly of directional operators in divergence form with conservative closure, (2) Kronecker lifting for 2D tensor grids, (3) Padé(0,2) factor application via banded or complex banded solves, (4) factorized Picard iteration for the mixed derivative block, (5) symmetric Strang composition of five substeps, (6) threshold computation via bisection on matrix function entries, (7) transpose duality for backward equation, (8) FFT-based jump operator application for translation-invariant kernels. The code is available on GitHub. The main complexity lies in correctly implementing the conservative closure, the factorized Picard solver, and the threshold certification. The scheme is linear algebra heavy but uses standard banded solvers.

The paper provides a GitHub repository with supporting code (https://github.com/rakhymzhan11/DF-ADI), detailed algorithmic descriptions, explicit threshold computation procedures via bisection, and comprehensive numerical benchmarks with specific parameters. All computations were performed in Matlab R2025b on Apple silicon. The companion paper [Itkin, 2026a] provides the spatial discretization details. However, some theoretical proofs are in appendices and the exact Picard iteration implementation details could benefit from more pseudocode.