The Euler scheme for Levy driven stochastic differential equations: limit theorems

We study the Euler scheme for a stochastic differential equation driven by a Levy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(X^n-X), where X is the true solution and X^n is its Euler approximation with stepsize 1/n, and u_n is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence (u_n) is a rate, which, in addition, is sharp when the limiting process (or processes) is not trivial. We suppose that Y has no Gaussian part (otherwise a rate is known to be u_n=\sqrt n). Then rates are given in terms of the concentration of the Levy measure of Y around 0 and, further, we prove the convergence of the sequence u_n(X^n-X) to a nontrivial limit under some further assumptions, which cover all stable processes and a lot of other Levy processes whose Levy measure behave like a stable Levy measure near the origin. For example, when Y is a symmetric stable process with index \alpha \in(0,2), a sharp rate is u_n=(n/\log n)^{1/\alpha}; when Y is stable but not symmetric, the rate is again u_n=(n/\log n)^{1/\alpha} when \alpha >1, but it becomes u_n=n/(\log n)^2 if \alpha =1 and u_n=n if \alpha <1.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000066

A test for the rank of the volatility process: the random perturbation approach

In this paper we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. Finally, we demonstrate a homoscedasticity test for the rank process.Comment: 30 page

Estimating the degree of activity of jumps in high frequency data

We define a generalized index of jump activity, propose estimators of that index for a discretely sampled process and derive the estimators' properties. These estimators are applicable despite the presence of Brownian volatility in the process, which makes it more challenging to infer the characteristics of the small, infinite activity jumps. When the method is applied to high frequency stock returns, we find evidence of infinitely active jumps in the data and estimate their index of activity.Comment: Published in at http://dx.doi.org/10.1214/08-AOS640 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Is Brownian motion necessary to model high-frequency data?

This paper considers the problem of testing for the presence of a continuous part in a semimartingale sampled at high frequency. We provide two tests, one where the null hypothesis is that a continuous component is present, the other where the continuous component is absent, and the model is then driven by a pure jump process. When applied to high-frequency individual stock data, both tests point toward the need to include a continuous component in the model.Comment: Published in at http://dx.doi.org/10.1214/09-AOS749 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fisher's Information for Discretely Sampled Levy Processes

This paper studies the asymptotic behavior of the Fisher information for a Levy process discretely sampled at an increasing frequency. We show that it is possible to distinguish not only the continuous part of the process from its jumps part, but also different types of jumps, and derive the rates of convergence of efficient estimators.Comment: 17 novembre 200

Limit theorems for moving averages of discretized processes plus noise

This paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency. Our method generalizes the pre-averaging approach (see [Bernoulli 15 (2009) 634--658, Stochastic Process. Appl. 119 (2009) 2249--2276]) and provides consistent estimates for various characteristics of general semimartingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate $n^{-1/4}$, if $n$ is the number of observations.Comment: Published in at http://dx.doi.org/10.1214/09-AOS756 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org