Paper: arXiv 2509.01041

Authors: Ziyao Wang, Svetlozar T Rachev

Abstract

Financial returns are known to exhibit heavy tails, volatility clustering and abrupt jumps that are poorly captured by classical diffusion models. Advances in machine learning have enabled highly flexible functional forms for conditional means and volatilities, yet few models deliver interpretable state–dependent tail risk, capture multiple forecast horizons and yield distributions amenable to backtesting and execution. This paper proposes a neural Lévy jump–diffusion framework that jointly learns, as functions of observable state variables, the conditional drift, diffusion, jump intensity and jump size distribution. We show how a single shared encoder yields multiple forecasting heads corresponding to distinct horizons (daily, weekly, etc.), facilitating multi–horizon density forecasts and risk measures. The state vector includes conventional price and volume features as well as novel complexity measures such as permutation entropy and recurrence quantification analysis determinism, which quantify predictability in the underlying process. Estimation is based on a quasi–maximum likelihood approach that separates diffusion and jump contributions via bipower variation weights and incorporates monotonicity and smoothness regularisation to ensure identifiability. A cost–aware portfolio optimiser translates the model’s conditional densities into implementable trading strategies under leverage, turnover and no–trade–band constraints. Extensive empirical analyses on cross–sectional equity data demonstrate improved calibration, sharper tail control and economically significant risk reduction relative to baseline diffusive and GARCH benchmarks. The proposed framework is therefore an interpretable, testable and practically deployable method for state–dependent risk and density forecasting.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly rigorous synthesis of neural SDEs, state-dependent jump-diffusion models, and dynamical complexity measures like permutation entropy and RQA determinism. The empirical framework is well-developed with multi-horizon density forecasting, quasi-maximum likelihood using bipower variation, and cost-aware portfolio optimization under real-world constraints.

Research Flowchart

  flowchart TD
    A["Research Goal: State-Dependent Risk & Density Forecasting<br><i>(Capture heavy tails, jumps & volatility clustering)</i>"] --> B["Inputs & Feature Engineering<br><i>(Price/volume + Entropy & RQA determinism features)</i>"]
    
    B --> C["Neural Lévy SDE Framework<br><i>(Shared encoder for drift, diffusion, jump intensity & size)</i>"]
    
    C --> D["Estimation & Regularization<br><i>(QMLE with bipower variation + monotonicity constraints)</i>"]
    
    D --> E["Multi-Horizon Forecasting & Portfolio Optimization<br><i>(Daily/weekly density heads + cost-aware trading constraints)</i>"]
    
    E --> F["Key Outcomes<br><i>(Superior calibration, sharper tail control & reduced risk vs GARCH)</i>"]