SigMA: Path Signatures and Multi-head Attention for Learning Parameters in fBm-driven SDEs

Stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) are increasingly used to model systems with rough dynamics and long-range dependence, such as those arising in quantitative finance and reliability engineering. However, these processes are non-Markovian and lack a semimartingale structure, rendering many classical parameter estimation techniques inapplicable or computationally intractable beyond very specific cases. This work investigates two central questions: (i) whether integrating path signatures into deep learning architectures can improve the trade-off between estimation accuracy and model complexity, and (ii) what constitutes an effective architecture for leveraging signatures as feature maps. We introduce SigMA (Signature Multi-head Attention), a neural architecture that integrates path signatures with multi-head self-attention, supported by a convolutional preprocessing layer and a multilayer perceptron for effective feature encoding. SigMA learns model parameters from synthetically generated paths of fBm-driven SDEs, including fractional Brownian motion, fractional Ornstein-Uhlenbeck, and rough Heston models, with a particular focus on estimating the Hurst parameter and on joint multi-parameter inference, and it generalizes robustly to unseen trajectories. Extensive experiments on synthetic data and two real-world datasets (i.e., equity-index realized volatility and Li-ion battery degradation) show that SigMA consistently outperforms CNN, LSTM, vanilla Transformer, and Deep Signature baselines in accuracy, robustness, and model compactness. These results demonstrate that combining signature transforms with attention-based architectures provides an effective and scalable framework for parameter inference in stochastic systems with rough or persistent temporal structure.

Filtered Markovian Projection: Dimensionality Reduction in Filtering for Stochastic Reaction Networks

Stochastic reaction networks (SRNs) model stochastic effects for various applications, including intracellular chemical or biological processes and epidemiology. A typical challenge in practical problems modeled by SRNs is that only a few state variables can be dynamically observed. Given the measurement trajectories, one can estimate the conditional probability distribution of unobserved (hidden) state variables by solving a stochastic filtering problem. In this setting, the conditional distribution evolves over time according to an extensive or potentially infinite-dimensional system of coupled ordinary differential equations with jumps, known as the filtering equation. The current numerical filtering techniques, such as the Filtered Finite State Projection (DAmbrosio et al., 2022), are hindered by the curse of dimensionality, significantly affecting their computational performance. To address these limitations, we propose to use a dimensionality reduction technique based on the Markovian projection (MP), initially introduced for forward problems (Ben Hammouda et al., 2024). In this work, we explore how to adapt the existing MP approach to the filtering problem and introduce a novel version of the MP, the Filtered MP, that guarantees the consistency of the resulting estimator. The novel method combines a particle filter with reduced variance and solving the filtering equations in a low-dimensional space, exploiting the advantages of both approaches. The analysis and empirical results highlight the superior computational efficiency of projection methods compared to the existing filtered finite state projection in the large dimensional setting.