Quantile estimation for Lévy measures

Abstract

Generalizing the concept of quantiles to the jump measure of a Lévy process, the generalized quantiles $q_{\tau }^{\pm }>0$, for $\tau >0$, are given by the smallest values such that a jump larger than $q_{\tau }^{+}$ or a negative jump smaller than $-q_{\tau }^{-}$, respectively, is expected only once in $1/\tau$ time units. Nonparametric estimators of the generalized quantiles are constructed using either discrete observations of the process or using option prices in an exponential Lévy model of asset prices. In both models minimax convergence rates are shown. Applying Lepski's approach, we derive adaptive quantile estimators. The performance of the estimation method is illustrated in simulations and with real data.

38 pages, 1 figure

References in corpus (8)

• Spectral estimation of the fractional order of a Lévy process
• Estimation for Lévy processes from high frequency data within a long time interval
• Asymptotic Equivalence for Nonparametric Regression
• Nonparametric inference on Lévy measures and copulas
• Confidence sets in nonparametric calibration of exponential Lévy models
• Option calibration of exponential Lévy models: Confidence intervals and empirical results
• Polar sets of anisotropic Gaussian random fields
• Adaptive nonparametric estimation for Lévy processes observed at low frequency

Cited by in corpus (2)

• Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme
• Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations