Graph structure learning for stable processes

TL;DR

The paper introduces Ising--Hüsler--Reiss (IHR) processes, a sparse, multivariate Lévy model that couples a Hüsler--Reiss dependence structure with Ising-based orthant weights to capture asymmetric jump dependencies across marginal stable processes. A novel uniform small-time approximation of the Lévy measure enables consistent, data-driven learning of both the conditional-independence graph $G$ (via a variogram-based reconstruction and EGlearn) and the Ising weights $\Psi$ on $G$, even when marginal stability indices differ. The authors demonstrate graph recovery and Ising-weight estimation through extensive simulations on Barabási–Albert graphs and apply the method to modeling dependence among stock returns, showing improved fit over tree-based approaches and meaningful interpretation of edgewise asymmetry. The framework provides a robust, scalable way to model and infer high-dimensional dependence among heavy-tailed, non-Gaussian processes with potential applications in finance and risk management.

Abstract

We introduce Ising-Hüsler-Reiss processes, a new class of multivariate Lévy 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 Lévy 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 Lévy processes. The methodology is illustrated in simulations and a real data application to modeling dependence of stock returns.

Figures (11)

• Figure 1: Simulation of a two-dimensional Ising--Hüsler--Reiss Lévy process with symmetric (left) and asymmetric (center) orthant weights. The corresponding jumps (right) of the Lévy measure are shown for both the symmetric (blue) and asymmetric (green) process.
• Figure 2: Boxplots of $F_1$-scores for estimation of tree graphs in the asymmetric regime for EGLearn based on neighborhood selection (NS) and graphical lasso (Glasso) for various penalty parameters and models selected by the two information criteria AIC and BIC, and the minimum spanning trees based on $\Gamma$ and $\chi$, for samples sizes $n=2000$ (top) and $n=10000$ (bottom) and dimensions $d=10$ (left) and $d=20$ (right).
• Figure 3: Boxplots of $F_1$-scores for estimation of Barabasi--Albert graphs with attachment parameter $a=2$ in the asymmetric regime for EGLearn based on neighborhood selection (NS) and graphical lasso (Glasso) for various penalty parameters and models selected by the two information criteria AIC and BIC, and the minimum spanning trees based on $\Gamma$ and $\chi$, for samples sizes $n=2000$ (top) and $n=10000$ (bottom) and dimensions $d=10$ (left) and $d=20$ (right).
• Figure 4: Boxplots of the parameter estimates of $\Psi$ along the edges of the graph $E$ for dimensions $d=5$ (top) and $d=10$ (bottem) and sample sizes $n=500$ (left), $n=2000$ (middle) and $n=10000$ (right). Displayed in ascending order of the true underlying parameter $\psi_{i,j}$ (blue).
• Figure 5: Estimated trees $\hat{T}^\chi$engelke2024levygraphicalmodels (left) and $\hat{T}^\Gamma$ (middle), and the estimated general graph $\hat{G}$ based on AIC (right). The colors encode the industry sectors of the underlying assets given by Financials (yellow), Consumer Staples (blue), Technology (purple), Energy (green).

Theorems & Definitions (50)

• Example 2.1
• Example 2.2
• Example 2.3
• Definition 3.1
• Lemma 3.1
• Corollary 3.1
• Example 3.1
• Theorem 3.1
• Remark 3.1
• Proposition 4.1