Generative Path-Law Jump-Diffusion: Sequential MMD-Gradient Flows and Generalisation Bounds in Marcus-Signature RKHS

Battle Count: 54

The paper is highly relevant to quantitative trading and finance. It addresses the critical challenge of generating realistic stochastic paths that incorporate structural breaks, regime shifts, and jump discontinuities—features essential for risk management, derivatives pricing, and strategy backtesting. The Marcus-signature framework provides a principled way to capture non-commutative higher-order moments (leverage effects, volatility clustering) that are crucial in financial markets. The jump-diffusion modeling directly addresses fat-tailed returns and black-swan events. However, the lack of empirical validation and the extreme theoretical complexity limit immediate practical applicability. The O(m²) computational scheme and Nyström compression make real-time implementation theoretically feasible.

The implementation complexity is extremely high. The framework requires: (1) computing time-extended Marcus-sense signatures on càdlàg paths in infinite-dimensional tensor algebras; (2) maintaining and updating a time-evolving precision operator Q_s via Sherman-Morrison-Woodbury rank-1 updates; (3) implementing a hybrid Euler-Maruyama-Marcus integration scheme with Poisson jump logic; (4) training a Neural CDE for latent state propagation; (5) implementing Nyström-compressed score-matching in signature RKHS; (6) managing the coupled jump-geometry system with dynamic spectral whitening. The mathematical prerequisites span rough path theory, stochastic analysis on Skorokhod spaces, functional analysis in RKHS, optimal transport, and statistical learning theory. No reference implementation or code is provided.

The paper is purely theoretical with no empirical experiments, no code repository, no datasets, and no numerical results. It presents mathematical definitions, propositions, theorems, proofs, and an algorithmic description (Algorithm 1), but provides no implementation details sufficient for reproduction. The framework relies on proprietary concepts from the author's prior working papers (Bloch 2025a, 2025b, 2026a, 2026b) which are also unpublished working papers. No open-source code or data is referenced.