Graph structure learning for stable processes

TL;DR

This paper introduces Ising-H"usler-Reiss processes, a new class of multivariate L"evy processes that enable sparse, data-driven modeling of complex dependence structures and asymmetries in stable processes, with proven estimators and practical applications.

Contribution

It proposes a novel class of processes with a flexible graphical structure and asymmetry modeling, along with consistent estimation methods and applications.

Findings

Consistent estimators for the graphical structure and asymmetry parameters.

Effective modeling of dependence and asymmetry in stock returns.

Validation through simulations and real data analysis.

Abstract

We introduce Ising-H\"usler-Reiss processes, a new class of multivariate L\'evy processes that allows for sparse modeling of the path-wise conditional independence structure between marginal stable processes with different stability indices. The underlying conditional independence graph is encoded as zeroes in a suitable precision matrix. An Ising-type parametrization of the weights for each orthant of the L\'evy measure allows for data-driven modeling of asymmetry of the jumps while retaining an arbitrary sparse graph. We develop consistent estimators for the graphical structure and asymmetry parameters, relying on a new uniform small-time approximation for L\'evy processes. The methodology is illustrated in simulations and a real data application to modeling dependence of stock returns.