Low Frequency Lévy Copula Estimation

Abstract

Let $X$ be a $d$-dimensional Lévy process with Lévy triplet $(\Sigma ,\nu ,\alpha )$ and $d\ge 2$. Given the low frequency observations $(X_t{)}_{t=1,\dots ,n}$, the dependence structure of the jumps of $X$ is estimated. The Lévy 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 Lévy 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 Reiß \cite{NeuRei} and Nickl and Reiß \cite{NicRei} to construct a Lévy copula estimator ${\hat{\mathfrak{C}}}_n$ based on the above $n$ observations. In doing so we prove $${\hat{\mathfrak{C}}}_n\to \mathfrak{C},n\to \infty$$ 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 Lévy triplets with $\Sigma \ne 0$ and $\nu$ of arbitrary Blumenthal-Getoor index $0\le \beta \le 2$. 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 ${\hat{C}}_n$ 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 Lévy copula $\mathfrak{C}$ in the finite jump activity case, i.e. the CPP case. Here we establish $${\hat{C}}_n\to C,n\to \infty$$ 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 Reiß.

References in corpus (1)

• Nonparametric inference on Lévy measures and copulas