Unveiling Long-Memory Dynamics in Turbulent Markets: A Novel Fractional-Order Attention-Based GRU-LSTM Framework with Multifractal Analysis

Financial time series in turbulent markets exhibit complex long-memory dynamics and multifractal features that traditional deep learning models fail to capture due to inherent exponential forgetting mechanisms. To address this, we propose Frac-Attn-GL, a novel Fractional-order Spatiotemporal Attention-based GRU-LSTM framework. Grounded in the Fractal Market Hypothesis, the model embeds Grünwald–Letnikov fractional-order operators into a dual-channel architecture (FracLSTM and FracGRU) to characterize long-range memory with rigorous power-law decay priors. Furthermore, an extreme-aware asymmetric loss function is designed to drive a dynamic spatiotemporal routing mechanism, enabling adaptive shifts between long-term macro trends and short-term micro shocks. Empirical tests on major U.S. stock indices reveal three significant findings. First, the Frac-Attn-GL framework substantially reduces prediction errors, achieving up to a 93.1% RMSE reduction on the highly volatile NASDAQ index compared to standard baselines. Second, the adaptively learned fractional-order parameters exhibit a consistent quantitative alignment with the market’s empirical multifractal singularity spectrum, supporting the physical interpretability of the model’s endogenous memory mechanism. Finally, hybrid residual multifractal diagnostics indicate that the framework effectively captures deep long-range correlations, reducing the Hurst exponent of the prediction residuals from ~0.83 to approximately 0.50, a level consistent with the absence of significant long-range dependence.