727 research outputs found
Quantile estimation for L\'evy measures
Generalizing the concept of quantiles to the jump measure of a L\'evy
process, the generalized quantiles , for , are given
by the smallest values such that a jump larger than or a
negative jump smaller than , respectively, is expected only once
in time units. Nonparametric estimators of the generalized quantiles
are constructed using either discrete observations of the process or using
option prices in an exponential L\'evy 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.Comment: 38 pages, 1 figur
Estimation of Volatility Functions in Jump Diffusions Using Truncated Bipower Increments
In the paper, we introduce and analyze a new methodology to estimate the volatility
functions of jump diffusion models. Our methodology relies on the standard kernel
estimation technique using truncated bipower increments. The relevant asymptotics
are fully developed, which allow for the time span to increase as well as the sampling
interval to decrease and accommodate both stationary and nonstationary recurrent
processes. We evaluate the performance of our estimators by simulation and provide
some illustrative empirical analyses
ABC of SV : limited information likelihood inference in stochastic volatility jump-diffusion models
We develop novel methods for estimation and filtering of continuous-time models with stochastic volatility and jumps using so-called Approximate Bayesian Computation which build likelihoods based on limited information. The proposed estimators and filters are computationally attractive relative to standard likelihood-based versions since they rely on low-dimensional auxiliary statistics and so avoid computation of high-dimensional integrals. Despite their computational simplicity, we find that estimators and filters perform well in practice and lead to precise estimates of model parameters and latent variables. We show how the methods can incorporate intra-daily information to improve on the estimation and filtering. In particular, the availability of realized volatility measures help us in learning about parameters and latent states. The method is employed in the estimation of a flexible stochastic volatility model for the dynamics of the S&P 500 equity index. We find evidence of the presence of a dynamic jump rate and in favor of a structural break in parameters at the time of the recent financial crisis. We find evidence that possible measurement error in log price is small and has little effect on parameter estimates. Smoothing shows that, recently, volatility and the jump rate have returned to the low levels of 2004-2006
Nonparametric inference for discretely sampled L\'evy processes
Given a sample from a discretely observed L\'evy process
of the finite jump activity, the problem of nonparametric estimation of the
L\'evy density corresponding to the process is studied. An estimator
of is proposed that is based on a suitable inversion of the
L\'evy-Khintchine formula and a plug-in device. The main results of the paper
deal with upper risk bounds for estimation of over suitable classes of
L\'evy triplets. The corresponding lower bounds are also discussed.Comment: 38 page
Detecting gradual changes in locally stationary processes
In a wide range of applications, the stochastic properties of the observed
time series change over time. The changes often occur gradually rather than
abruptly: the properties are (approximately) constant for some time and then
slowly start to change. In many cases, it is of interest to locate the time
point where the properties start to vary. In contrast to the analysis of abrupt
changes, methods for detecting smooth or gradual change points are less
developed and often require strong parametric assumptions. In this paper, we
develop a fully nonparametric method to estimate a smooth change point in a
locally stationary framework. We set up a general procedure which allows us to
deal with a wide variety of stochastic properties including the mean,
(auto)covariances and higher moments. The theoretical part of the paper
establishes the convergence rate of the new estimator. In addition, we examine
its finite sample performance by means of a simulation study and illustrate the
methodology by two applications to financial return data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1297 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes
A piecewise-deterministic Markov process is a stochastic process whose
behavior is governed by an ordinary differential equation punctuated by random
jumps occurring at random times. We focus on the nonparametric estimation
problem of the jump rate for such a stochastic model observed within a long
time interval under an ergodicity condition. We introduce an uncountable class
(indexed by the deterministic flow) of recursive kernel estimates of the jump
rate and we establish their strong pointwise consistency as well as their
asymptotic normality. We propose to choose among this class the estimator with
the minimal variance, which is unfortunately unknown and thus remains to be
estimated. We also discuss the choice of the bandwidth parameters by
cross-validation methods.Comment: 36 pages, 18 figure
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