1,668 research outputs found
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
, if 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
- …