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

Xianglin Wu, Chiheb Ben Hammouda and Cornelis W. Oosterlee

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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.

Date: 2025-12
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