SigMA: Neural Inference for fBm-driven SDEs

• SigMA is a neural architecture combining convolutional preprocessing, truncated path signatures, multi-head self-attention, and an MLP for parameter estimation in fBm-driven SDEs.
• It efficiently captures long-range dependencies and rough path structures, enabling accurate estimation of the Hurst parameter and joint inference of drift and volatility parameters.
• Empirical evaluations demonstrate that SigMA outperforms traditional models like CNNs, LSTMs, and Transformers in accuracy, robustness, and model compactness on synthetic and real-world datasets.

SigMA is a neural architecture for parameter inference in stochastic differential equations driven by fractional Brownian motion, designed for settings in which long-range dependence and rough temporal structure make classical estimation difficult or inapplicable. It combines path signatures with multi-head self-attention, together with a convolutional preprocessing layer and a multilayer perceptron, to learn model parameters from observed trajectories. The framework is evaluated on synthetically generated paths of fractional Brownian motion, fractional Ornstein–Uhlenbeck, and rough Heston models, with particular emphasis on estimation of the Hurst parameter and joint multi-parameter inference, and it is reported to generalize robustly to unseen trajectories (Wu et al., 17 Dec 2025).

1. Problem setting and statistical context

SigMA is situated in the estimation problem for SDEs of the form

$$dX(t) = \mu(X(t), \theta)\, dt + \sigma(X(t), \theta)\, dB^H(t),$$

where $B^H$ denotes fractional Brownian motion and $\theta$ is an unknown parameter vector (Wu et al., 17 Dec 2025). The paper emphasizes two structural obstacles. First, fBm-driven systems are non-Markovian. Second, they lack a semimartingale structure. In the formulation given for the model, these properties render many classical parameter estimation techniques inapplicable or computationally intractable beyond very specific cases (Wu et al., 17 Dec 2025).

Within this setting, the Hurst parameter $H$ is a central object because it controls memory and path roughness. The framework is also designed for joint estimation problems, including combinations of $H$, volatility-related parameters, and mean-reversion parameters in models such as fractional Ornstein–Uhlenbeck and rough Heston (Wu et al., 17 Dec 2025). The article’s motivating question is not merely whether a neural estimator can replace a classical one, but whether path signatures can improve the trade-off between estimation accuracy and model complexity, and what architecture best exploits them (Wu et al., 17 Dec 2025).

This places SigMA in a line of work that treats parameter estimation as supervised learning on simulated path data. In the formulation used here, a model $\mathcal{M}$ learns a map

$$\mathcal{M}: \mathbb{R}^{n \times d} \to \mathbb{R}^{d_\theta}$$

from a discretized path to parameter estimates by minimizing a training loss over synthetically generated trajectories with known ground-truth parameters (Wu et al., 17 Dec 2025). A plausible implication is that the method is most naturally suited to regimes in which a high-fidelity simulator is available even when closed-form statistical procedures are not.

2. Architectural composition

SigMA consists of four principal modules: a convolutional preprocessing layer, a path-signature transform, a multi-head self-attention block, and an MLP prediction head (Wu et al., 17 Dec 2025). The architecture is explicitly designed so that signatures provide a compact path representation while attention aggregates information across signature levels.

Component	Function	Formulation or note
Convolutional preprocessing	Extracts local features and denoises input paths	$$\Phi(\mathbf{X}^{(n)}; W^\Phi, B^\Phi) = W^\Phi \star \mathbf{X}^{(n)} + \mathbf{1}_{\tilde n} B^\Phi$$
Path signature transform	Encodes iterated path information up to order $N$	Truncated signature with time augmentation
Multi-head self-attention	Aggregates signature features across levels	Heads typically aligned with signature levels
MLP head	Produces final parameter estimates	Reported as 5 hidden layers of size 32 with ReLU

The input is a discretized path

$$\mathbf{X}^{(n)} = (X_0, X_1, \ldots, X_{n-1}) \in \mathbb{R}^{n \times d}.$$

The convolutional front end computes

$B^H$0

where $B^H$1 is the convolutional kernel, $B^H$2 is a bias term, and $B^H$3 denotes cross-correlation (Wu et al., 17 Dec 2025). The details state that this stage extracts local features and denoises the input path before higher-level feature extraction.

After convolution, time augmentation is applied before signature computation for uniqueness, and the sequence is lifted into overlapping subpaths. The paper gives a representative lifting operation as

$B^H$4

typically with stride $B^H$5 (Wu et al., 17 Dec 2025). This suggests that SigMA does not treat the entire path as a single indivisible object; rather, it constructs a structured set of local-to-global signature summaries.

The attention block then operates on the resulting matrix of signature features. The MLP maps the aggregated representation to parameter estimates. The paper reports typical hyperparameters of convolution with channels $B^H$6, kernel size $B^H$7, signature truncation $B^H$8, and stride $B^H$9 (Wu et al., 17 Dec 2025).

3. Path signatures as feature maps

The core representational device in SigMA is the truncated path signature. For a path $\theta$0, the signature to order $\theta$1 is defined as

$\theta$2

with

$\theta$3

The paper states two key properties: uniqueness for time-augmented paths up to translation, and factorial decay of signature terms (Wu et al., 17 Dec 2025).

Within SigMA, the signature is not used as a stand-alone handcrafted statistic. It is integrated into a trainable pipeline after convolutional preprocessing and before attention-based aggregation. The resulting representation is a matrix stacking the signatures of lifted path segments across truncation levels (Wu et al., 17 Dec 2025). This design is central to the claim that model complexity is decoupled from path length: the paper states that SigMA’s parameter count does not increase with path length due to signature truncation (Wu et al., 17 Dec 2025).

This contrasts with recurrent and transformer-style sequence baselines whose parameterization or effective computation is more directly tied to long sequential inputs. The article also notes that Deep Signature baselines perform well, which highlights the importance of path signatures themselves, not only of the downstream attention block (Wu et al., 17 Dec 2025). At the same time, single-head attention variants are reported to be less effective for complex or joint parameter estimation, which the paper presents as evidence that multi-head aggregation over signature structure is materially useful (Wu et al., 17 Dec 2025).

4. Attention mechanism and learning objective

The attention module applies self-attention to the matrix of signature features. In the formulation given in the paper, a single attention head is

$\theta$4

where $\theta$5, $\theta$6, and $\theta$7 are linear projections of the signature-feature matrix $\theta$8 (Wu et al., 17 Dec 2025). The multi-head version allows different heads to focus on different aspects of the path, and the details state that each head corresponds to a signature level (Wu et al., 17 Dec 2025).

The model is trained by supervised learning on synthetic trajectories. Given observed data $\theta$9 generated from an fBm-driven SDE with parameter vector $H$0, SigMA learns model parameters $H$1 by minimizing

$H$2

where $H$3 is task-dependent, such as RMSE or cross-entropy (Wu et al., 17 Dec 2025).

The paper describes the prediction head as an MLP with five hidden layers of width 32 and ReLU activations, followed by a final sigmoid or suitable task-specific activation (Wu et al., 17 Dec 2025). This setup yields an estimator that combines local feature extraction, rough-path-style summary statistics, cross-level attention, and standard feed-forward regression. A plausible implication is that the architecture is explicitly biased toward structured stochastic processes rather than generic time-series pattern recognition.

5. Empirical evaluation

The evaluation covers both synthetic and real-world data. Synthetic experiments are performed on fractional Brownian motion, fractional Ornstein–Uhlenbeck, and rough Heston models (Wu et al., 17 Dec 2025). Real-world evaluations use equity-index realized volatility data and Li-ion battery degradation data (Wu et al., 17 Dec 2025). The paper compares SigMA against CNN, LSTM, vanilla Transformer, and Deep Signature baselines.

The reported empirical pattern is consistent across several axes. For single-parameter estimation, especially the Hurst parameter, SigMA achieves either the best or near-best RMSE across the studied model classes (Wu et al., 17 Dec 2025). For multi-parameter estimation, it is reported to significantly outperform CNN, LSTM, Transformer, and DeepSigNet baselines in joint estimation of $H$4, drift, and volatility-related parameters, with lower RMSE and more robust relative squared error distributions (Wu et al., 17 Dec 2025). On the two real-world datasets, the model is reported to produce more accurate and stable Hurst estimates and to match or surpass statistical benchmarks such as R/S and Higuchi in robustness to trends and nonstationarity (Wu et al., 17 Dec 2025).

The paper also emphasizes compactness. SigMA is described as consistently outperforming CNN, LSTM, vanilla Transformer, and Deep Signature baselines in accuracy, robustness, and model compactness (Wu et al., 17 Dec 2025). Because signature truncation fixes representational order independently of observation length, the architecture avoids growth of parameter count with path length (Wu et al., 17 Dec 2025). This is presented as one of its principal scalability advantages.

Ablation studies further attribute performance to the combination of components rather than to a single module. Removing the convolutional preprocessing layer or the MLP degrades performance or stability, and replacing multi-head attention with a single-head variant reduces effectiveness for more complex inference tasks (Wu et al., 17 Dec 2025). This suggests that the model’s gain is not reducible to signatures alone, even though signatures are an essential part of the improvement.

6. Interpretation and position within rough-path-based learning

SigMA can be read as an overview of rough-path-inspired representation and modern attention-based sequence modeling. Its novelty, as stated in the paper, lies in integrating signature transforms with multi-head self-attention in a way that improves the trade-off between estimation accuracy and model complexity for fBm-driven SDEs (Wu et al., 17 Dec 2025). The architecture is thus neither a purely geometric signature model nor a generic transformer adapted to stochastic data.

A common misconception would be to treat SigMA as simply another temporal deep network for forecasting. The formulation in the paper is instead explicitly inferential: the target is parameter recovery for a generative stochastic system, not prediction of the next observation (Wu et al., 17 Dec 2025). Another plausible misconception is that signatures by themselves resolve the estimation problem. The reported ablations indicate that the combination of convolutional preprocessing, signatures, multi-head attention, and MLP refinement is materially stronger than reduced variants (Wu et al., 17 Dec 2025).

The paper frames the practical significance in application domains where roughness and persistence are intrinsic, including quantitative finance and reliability engineering (Wu et al., 17 Dec 2025). Its real-data experiments on realized volatility and battery degradation instantiate those use cases directly. More broadly, the reported ability to generalize robustly to unseen trajectories suggests that the framework is intended as a learned surrogate for difficult inverse problems in non-Markovian stochastic systems rather than as an ad hoc benchmark model (Wu et al., 17 Dec 2025).