Bayesian Non-Parametric Inference for Lévy Measures in State-Space Models

2025/05/28 by Bill Yuchen Lin, Lin, Bill Z., Simon Godsill +1
Computer Science · Decision Sciences · Economics, Econometrics and Finance · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #Financial Risk and Volatility Modeling #Methodology (stat.ME) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2505.22587

openalex publication_date 2025/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Lévy processes, known for their ability to model complex dynamics with skewness, heavy tails, and discontinuities, play a critical role in stochastic modeling across various domains. However, inference for most Lévy processes, whether in parametric or non-parametric settings, remains a significant challenge. In this work, we present a novel Bayesian non-parametric inference framework for inferring the Lévy measures of subordinators and normal variance-mean (NVM) processes within a linear state space model. A flexible random measure, the Independent Gamma-scaled Dirichlet Process (IGSDP), is introduced, for which the well-known Gamma process is a special case, leading to tractable conditional distributions for inference about both Lévy measures. We further show that in the Gamma process special case, conjugacy can be achieved for hyper-parameter inference. An explicit characterization of the parameter contour for NVM processes is provided, enabling an identifiable parameterization of the model for effective Markov Chain Monte Carlo algorithms in posterior inference. The method is demonstrated on both synthetic and tick-level (high-frequency) financial datasets.

Citations

• Nonparametric density estimation for the small jumps of Lévy processes
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• The Variance Gamma (V.G.) Model for Share Market Returns

Related

• The Lévy State Space Model
• Nonparametric estimation for Levy processes with a view towards mathematical finance
• Unbiased Parameter Inference for a Class of Partially Observed Levy-Process Models
• Generalised shot noise representations of stochastic systems driven by non-Gaussian Lévy processes
• A Bayesian Nonparametric approach to Reconstruction and Prediction of\n Random Dynamical Systems