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

Daniel Bloch

stat.MLApr 6, 2026

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

Authors: Daniel Bloch

Organizations: UNIVERSITY OF PARIS 6 & VINUNIVERSITY · Visting Professor at the College of Engineering and Computer Science, VinUniversity, Hanoi.

Abstract

This paper introduces a novel generative framework for synthesising forward-looking, càdlàg stochastic trajectories that are sequentially consistent with time-evolving path-law proxies, thereby incorporating anticipated structural breaks, regime shifts, and non-autonomous dynamics. By framing path synthesis as a sequential matching problem on restricted Skorokhod manifolds, we develop the Anticipatory Neural Jump-Diffusion (ANJD) flow, a generative mechanism that effectively inverts the time-extended Marcus-sense signature. Central to this approach is the Anticipatory Variance-Normalised Signature Geometry (AVNSG), a time-evolving precision operator that performs dynamic spectral whitening on the signature manifold to ensure contractivity during volatile regime shifts and discrete aleatoric shocks. We provide a rigorous theoretical analysis demonstrating that the joint generative flow constitutes an infinitesimal steepest descent direction for the Maximum Mean Discrepancy functional relative to a moving target proxy. Furthermore, we establish statistical generalisation bounds within the restricted path-space and analyse the Rademacher complexity of the whitened signature functionals to characterise the expressive power of the model under heavy-tailed innovations. The framework is implemented via a scalable numerical scheme involving Nyström-compressed score-matching and an anticipatory hybrid Euler-Maruyama-Marcus integration scheme. Our results demonstrate that the proposed method captures the non-commutative moments and high-order stochastic texture of complex, discontinuous path-laws with high computational efficiency.

Figures & tables

X_{s+h}^{(i)}=X_{s}^{(i)}+\underbrace{f_{\theta}(s,X_{s}^{(i)},\hat{\Phi}_{s|t})h}_{\text{Continuous Drift}}+\underbrace{g_{\theta}\Delta W_{s}^{(i)}}_{\text{Diffusion}}+\underbrace{h_{\theta}(s,X_{s}^{(i)},\hat{\Phi}_{s|t})\Delta N_{s}^{(i)}}_{\text{Marcus Jump}}

(6.18)

Algorithm 1 Flexible Anticipatory Jump-Diffusion Sampling (ANJD)

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