Bayesian Non-Parametric Inference for L\'evy Measures in State-Space Models

TL;DR

This paper introduces a Bayesian non-parametric framework using Dirichlet processes and MCMC for inferring Le9vy measures in state-space models, demonstrated on financial data for improved forecasting.

Contribution

It extends existing methodologies by providing a novel Bayesian non-parametric inference approach for Le9vy measures in linear state-space models, with an efficient MCMC algorithm.

Findings

Effective inference on synthetic data

Successful application to high-frequency financial data

Improved forecasting performance

Abstract

L\'evy 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 L\'evy 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 L\'evy measures of subordinators and normal variance-mean (NVM) processes within a linear L\'evy state space model, a setup that significantly extends existing methodologies. We employ the Dirichlet process which further results in a Student-t mixture representation to enable inference for the L\'evy measures. An efficient augmented Markov Chain Monte Carlo algorithm is developed for this problem that ensures both accuracy and computational feasibility. The effectiveness of the method is demonstrated on synthetic and tick-level (high-frequency) financial datasets, and we show the practical utility of the inference results using the forecasting performance.