Paper: arXiv 2604.05008

Authors: Daniel Bloch

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 \textit{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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents an extremely sophisticated theoretical framework combining Marcus signatures, MMD-gradient flows, Skorokhod manifolds, and Rademacher complexity bounds. However, it leans almost entirely on advanced mathematical formalisms with minimal concrete empirical backtesting or real-world data validation presented in the excerpt.

Research Flowchart

  flowchart TD
    A["<b>Research Goal:</b> Synthesise forward-looking, càdlàg paths matching time-evolving path-laws with regime shifts"] --> B["<b>Inputs & Geometry:</b> Path-law proxies, Marcus-signatures & Anticipatory Variance-Normalised Geometry (AVNSG)"]
    
    B --> C["<b>Methodology:</b> Anticipatory Neural Jump-Diffusion (ANJD) via Marcus-sense signature inversion"]
    
    C --> D["<b>Computation:</b> Nyström-compressed score-matching & Hybrid Euler-Maruyama-Marcus integration"]
    
    D --> E["<b>Theoretical Validation:</b> MMD-gradient flow proof & Rademacher generalisation bounds"]
    
    E --> F["<b>Outcomes:</b> Scalable generation of non-commutative moments & high-order stochastic path textures"]