Low Frequency L\'evy Copula Estimation

TL;DR

This paper develops a method to estimate the dependence structure of jumps in a multivariate Lévy process from low frequency data, achieving different convergence rates for general and compound Poisson processes.

Contribution

It introduces a new estimator for the Lévy copula based on low frequency observations and proves its convergence with optimal rates under broad conditions.

Findings

Estimator converges uniformly on compact sets away from zero.

Convergence rate is log n for general Lévy processes.

Convergence rate is n for compound Poisson processes.

Abstract

Let $X$ be a $d$-dimensional L\'evy process with L\'evy triplet $(\Sigma,\nu,\alpha)$ and $d\geq 2$. Given the low frequency observations $(X_t)_{t=1,\ldots,n}$, the dependence structure of the jumps of $X$ is estimated. The L\'evy measure $\nu$ describes the average jump behavior in a time unit. Thus, the aim is to estimate the dependence structure of $\nu$ by estimating the L\'evy copula $\mathfrak{C}$ of $\nu$, cf. Kallsen and Tankov \cite{KalTan}. We use the low frequency techniques presented in a one dimensional setting in Neumann and Rei{\ss} \cite{NeuRei} and Nickl and Rei{\ss} \cite{NicRei} to construct a L\'evy copula estimator $\widehat{\mathfrak{C}}_n$ based on the above $n$ observations. In doing so we prove $$\widehat{\mathfrak{C}}_n\to \mathfrak{C},\quad n\to\infty$$ uniformly on compact sets bounded away from zero with the convergence rate $\sqrt{\log n}$. This convergence holds under quite general assumptions, which also include L\'evy triplets with $\Sigma\neq 0$ and $\nu$ of arbitrary Blumenthal-Getoor index $0\leq\beta\leq 2$. Note that in a low frequency observation scheme, it is statistically difficult to distinguish between infinitely many small jumps and a Brownian motion part. Hence, the rather slow convergence rate $\sqrt{\log n}$ is not surprising. In the complementary case of a compound Poisson process (CPP), an estimator $\widehat{C}_n$ for the copula $C$ of the jump distribution of the CPP is constructed under the same observation scheme. This copula $C$ is the analogue to the L\'evy copula $\mathfrak{C}$ in the finite jump activity case, i.e. the CPP case. Here we establish $$\widehat{C}_n \to C,\quad n\to\infty$$ with the convergence rate $\sqrt{n}$ uniformly on compact sets bounded away from zero. Both convergence rates are optimal in the sense of Neumann and Rei{\ss}.