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

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.

• The paper introduces a generative framework leveraging time-extended Marcus-signatures and ANJD flows to capture jump-diffusions and structural shocks.
• It employs MMD gradient flows and adaptive spectral whitening (AVNSG) to stabilize and accurately model discontinuous stochastic dynamics.
• Generalisation bounds and error decays, with an O(1/√n) rate, confirm the method's robustness in high-dimensional, irregular environments.

Generative Path-Law Jump-Diffusion: Deep Signature Geometry, MMD Gradient Flows, and Generalisation in Skorokhod Manifolds

Introduction and Context

This paper introduces a generative framework tailored to forward-looking path synthesis in environments characterized by discontinuous stochastic processes, regime shifts, and heteroskedastic volatility—ubiquitous challenges in quantitative finance and stochastic modeling. The approach leverages the algebraic structure provided by time-extended signatures in the Marcus sense to encode the full law of jump-diffusions, effectively mapping the path-measure into a reproducing kernel Hilbert space (RKHS) suitable for variational optimization. The authors propose Anticipatory Neural Jump-Diffusion (ANJD) flows, utilizing anticipatory, variance-normalised signature geometry (AVNSG) to ensure the contractive evolution of the generative flow along restricted Skorokhod manifolds.

Theoretical Innovations

Time-Extended Marcus-Sense Signature Geometry

At the core of the framework is the Marcus-sense signature, which encodes the non-commutative moments of paths including jump-discontinuities (as opposed to only continuous rough paths). By incorporating a time-extended component in the signature, injectivity within the Skorokhod space is achieved, ensuring that the mean embedding $\Phi_\mu = \mathbb{E}_{\mu}[S(\tilde{\gamma})]$ is a characteristic statistic for the path-law of any jump-diffusion, including its discrete shocks and regime switches.

Anticipatory Neural Jump-Diffusion (ANJD) Architecture

The ANJD process models path evolution as a system of neural jump-SDEs, with drift, diffusion, and jump intensity parameterized by the latent signature proxy $\hat{\Phi}_{s|t}$ and an adaptive geometry $\mathcal{Q}_s$. The latent proxy is propagated forward by a controlled differential equation (neural CDE) with the forward path extension $\hat{X}$ serving as the control. Through this, anticipated structural breaks and volatility spikes are natively integrated into the generative law.

Maximum Mean Discrepancy (MMD) Gradient Flows

A key theoretical result establishes that ANJD flows constitute an infinitesimal steepest descent in the MMD functional defined in the Marcus-signature RKHS. This underpins the coupling between the stochastic generator's drift/jump parameters and the stochastic geometry of the target law, aligning the expected Marcus-signature of the generative process with the time-evolving latent proxy. The endogenous jump intensities $\lambda_\theta$ and amplitudes $h_\theta$ produce discrete mass transport across signature space, crucial for accurate modeling of structural shocks.

Spectral Whitening via AVNSG

The adaptive geometry regularizes the signature manifold using a precision operator $\mathcal{Q}_s = (\Omega_s + \lambda I)^{-1}$, where $\Omega_s$ is the long-run signature covariance. AVNSG performs dynamic spectral whitening, ensuring stability and contractivity of the generative flow even during anticipated volatility explosions and in the presence of heavy-tailed innovations.

Statistical Guarantees and Complexity Analysis

The authors derive high-probability generalisation bounds quantifying the discrepancy between the (infinite-dimensional) theoretical signature embedding and its empirical realization in AVNSG. The generalisation rate depends on the Rademacher complexity of the whitened signature manifold:

• Letting $$R_s = \sup_{\gamma \in \text{supp}(\mu_s)} \|S(\tilde{\gamma})\|_{\mathcal{Q}_s}$$, the generalisation error decays as $O(1/\sqrt{n})$ up to deterministic factors parameterized by $\hat{\Phi}_{s|t}$0. The spectral regularisation of $\hat{\Phi}_{s|t}$1 attenuates the influence of high-order tensor components introduced by jumps and time-drift, stabilizing $\hat{\Phi}_{s|t}$2 and preventing kurtosis-driven divergence.
• The empirical Rademacher complexity of the function class $\hat{\Phi}_{s|t}$3 in $\hat{\Phi}_{s|t}$4 is explicitly bounded by the spectral alignment of Nyström-projected signature features with the principal eigenspaces of $\hat{\Phi}_{s|t}$5, ensuring that "black-swan" jump effects do not exponentially increase sample complexity.

Implementation and Numerical Scheme

Real-time feasibility is achieved by projecting the infinite-dimensional signature dynamics onto principal Nyström subspaces, with computational cost $\hat{\Phi}_{s|t}$6 per update. The jump-aware Sherman-Morrison-Woodbury formula enables fast rank-1 updates to the precision operator upon path extension or structural breaks. The generative scheme integrates both the continuous and jump-driven components via a hybrid Euler-Maruyama-Marcus (EMM) step, ensuring proper Marcus integration across discontinuities.

Score-matching aligns the generator's infinitesimal dynamics to the moving target's signature velocity, approximated via precision-weighted residuals in the signature manifold. The resulting framework supports both out-of-sample simulation (forecasting) and in-sample reconstruction (structural interpolation) seamlessly.

Implications, Strengths, and Future Directions

This work provides a blueprint for structure-preserving generative modeling of non-stationary stochastic processes with discontinuities—capabilities that standard diffusion models and neural SDEs lack due to insufficient anchoring to the path-law's non-commutative moments. By leveraging time-extended Marcus signatures, the method ensures strong universal approximation properties and uniquely identifies the law of the process in the presence of jumps.

The statistical results demonstrate that, under AVNSG, the generalisation and approximation errors are fundamentally regularised by the spectral properties of the latent geometry rather than by the tail-heaviness of the underlying measure. This yields robustness to extreme regime switches and supports practical deployment in finance and other domains where catastrophic shocks dominate risk.

Future research is likely to extend these flows to multi-agent systems and high-dimensional coupled jump-diffusions, integrate deep signature learning architectures for complex decision-making under uncertainty, and further develop real-time risk management pipelines for discontinuous stochastic environments.

Conclusion

The "Generative Path-Law Jump-Diffusion" framework advances the theory and practice of generative modeling for irregular stochastic environments via an overview of signature geometry, adaptive spectral regularisation, and steepest-descent transport in path-space. It establishes the Marcus-signature as a computational and statistical backbone for universal representation and robust synthesis of path-laws containing structural discontinuities. The resulting methodology enables real-time, structure-aware synthesis of stochastic processes critical for simulation, filtering, and scenario generation in regimes dominated by non-stationary shocks and time-varying dynamics (2604.05008).