The L\'evy State Space Model

TL;DR

This paper introduces a new class of Le9vy-driven state space models based on shot-noise representations, capable of capturing heavy-tailed non-Gaussianity while allowing for tractable inference, especially for irregular data.

Contribution

It proposes a novel Le9vy state space modeling framework that handles non-Gaussian heavy tails and enables parameter marginalization for improved inference.

Findings

Successfully applied Rao-Blackwellised SMC to real exchange rate data.

Models can marginalize skewness and scale parameters from posterior distributions.

Handles irregular data arrival times effectively.

Abstract

In this paper we introduce a new class of state space models based on shot-noise simulation representations of non-Gaussian L\'evy-driven linear systems, represented as stochastic differential equations. In particular a conditionally Gaussian version of the models is proposed that is able to capture heavy-tailed non-Gaussianity while retaining tractability for inference procedures. We focus on a canonical class of such processes, the $\alpha$-stable L\'evy processes, which retain important properties such as self-similarity and heavy-tails, while emphasizing that broader classes of non-Gaussian L\'evy processes may be handled by similar methodology. An important feature is that we are able to marginalise both the skewness and the scale parameters of these challenging models from posterior probability distributions. The models are posed in continuous time and so are able to deal with irregular data arrival times. Example modelling and inference procedures are provided using Rao-Blackwellised sequential Monte Carlo applied to a two-dimensional Langevin model, and this is tested on real exchange rate data.