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

Paper: arXiv 2512.15088

Authors: Xianglin Wu, Chiheb Ben Hammouda, Cornelis W. Oosterlee

Abstract

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.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a novel fusion of path signatures and multi-head self-attention mechanisms to solve challenging parameter estimation tasks in non-Markovian, fBm-driven SDEs. It combines strong analytical foundations (rough path theory and fractional processes) with thorough empirical benchmarking across synthetic and real-world financial and engineering datasets.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br/>Parameter estimation in non-Markovian,<br/>fBm-driven SDEs (e.g., Hurst parameter H)"] --> B["<b>Data Sources</b><br/>Synthetic SDE paths (fBm, fOU, rough Heston)<br/>& Real Data (Volatility, Battery Degradation)"]
    
    B --> C["<b>Signature Encoding</b><br/>Extract path signatures to capture<br/>rough dynamics & non-Markovian features"]
    
    C --> D["<b>SigMA Architecture</b><br/>Process features via 1D-CNN,<br/>Multi-Head Self-Attention, & MLP"]
    
    D --> E["<b>Training & Evaluation</b><br/>Train on synthetic paths;<br/>Test joint parameter inference & robustness"]
    
    E --> F["<b>Key Outcomes</b><br/>SigMA outperforms CNN, LSTM, & Transformers;<br/>Achieves superior accuracy with fewer parameters"]

Cited References (33)

1. Uniqueness for the signature of a path of bounded variation and the reduced path group (2010)
2. Deep signature transforms (2019)
3. Differential equations driven by rough signals (1998)
4. A fast algorithm for simulation of rough volatility models (2022)
5. Fractional Brownian motions, fractional noises and applications (1968)
6. Calibrating rough volatility models: a convolutional neural network approach (2020)
7. SigFormer: Signature Transformers for Deep Hedging (2023)
8. Attention is all you need (2017)

0 of 33 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.