Searcharxiv⌕ Search

arXiv subjects

Mark Podolskij

Publications and source records attributed to Mark Podolskij.

At least 55 records · Page 3Linked to original sources
arXiv2018

Estimation of the linear fractional stable motion

In this paper we investigate the parametric inference for the linear fractional stable motion in high and low frequency setting. The symmetric linear fractional stable motion is a three-parameter family, which constitutes a natural non-Gaussian analogue of the scaled fractional Brownian motion. It is fully characterised by the scaling parameter $σ>0$, the self-similarity parameter $H \in (0,1)$ and the stability index $α\in (0,2)$ of the driving stable motion. The parametric estimation of the model is inspired by the limit theory for stationary increments Lévy moving average processes that has been recently studied in \cite{BLP}. More specifically, we combine (negative) power variation statistics and empirical characteristic functions to obtain consistent estimates of $(σ, α, H)$. We present the law of large numbers and some fully feasible weak limit theorems.

stat.ME↗
arXiv2017

Asymptotic behavior of local times related statistics for fractional Brownian motion

We consider high frequency observations from a fractional Brownian motion. Inspired by the work of Jean Jacod in a diffusion setting, we investigate the asymptotic behavior of various classical statistics related to the local times of the process. We show that as in the diffusion case, these statistics indeed converge to some local times up to a constant factor. As a corollary, we provide limit theorems for the quadratic variation of the absolute value of a fractional Brownian motion.

math.PR↗
arXiv2016

On limit theory for Levy semi-stationary processes

In this paper we present some limit theorems for power variation of Lévy semi-stationary processes in the setting of infill asymptotics. Lévy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving average type processes with a multiplicative random component, which is usually referred to as volatility or intermittency. From the mathematical point of view this work extends the asymptotic theory investigated in [14], where the authors derived the limit theory for $k$th order increments of stationary increments Lévy driven moving averages. The asymptotic results turn out to heavily depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal--Getoor index $β\in (0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $α$. In this paper we will study the first order asymptotic theory for Lévy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.

math.PR↗
arXiv2016

Estimation of the global regularity of a multifractional Brownian motion

This paper presents a new estimator of the global regularity index of a multifractional Brownian motion. Our estimation method is based upon a ratio statistic, which compares the realized global quadratic variation of a multifractional Brownian motion at two different frequencies. We show that a logarithmic transformation of this statistic converges in probability to the minimum of the Hurst function, which is, under weak assumptions, identical to the global regularity index of the path.

math.PR↗
arXiv2016

Limit theorems for a class of stationary increments Levy driven moving averages

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal--Getoor index $β\in (0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $α$. First order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove the second order limit theorem connected to the ergodic type result. When the driving Lévy process $L$ is a symmetric $β$-stable process we obtain two different limits: a central limit theorem and convergence in distribution towards a $(1-α)β$-stable totally right skewed random variable.

math.PR↗
arXiv2015

Edgeworth expansion for the pre-averaging estimator

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.

math.ST↗
arXiv2015

Limit theorems for stationary increments Lévy driven moving averages

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy driven moving averages. In this infill sampling setting, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal-Getoor index $β\in (0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $α$. First order asymptotic theory essentially comprise three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove the second order limit theorem connected to the ergodic type result. When the driving Lévy process $L$ is a symmetric $β$-stable process we obtain two different limits: a central limit theorem and convergence in distribution towards a $(1-α)β$-stable random variable.

math.PR↗
arXiv2015

On U- and V-statistics for discontinuous Ito semimartingales

In this paper we examine the asymptotic theory for U-statistics and V-statistics of discontinuous Ito semimartingales that are observed at high frequency. For different types of kernel functions we show laws of large numbers and associated stable central limit theorems. In most of the cases the limiting process will be conditionally centered Gaussian. The structure of the kernel function determines whether the jump and/or the continuous part of the semimartingale contribute to the limit.

math.PR↗
arXiv2014

On spectral distribution of high dimensional covariation matrices

In this paper we present the asymptotic theory for spectral distributions of high dimensional covariation matrices of Brownian diffusions. More specifically, we consider $N$-dimensional Ito integrals with time varying matrix-valued integrands. We observe $n$ equidistant high frequency data points of the underlying Brownian diffusion and we assume that $N/n\rightarrow c\in (0,\infty)$. We show that under a certain mixed spectral moment condition the spectral distribution of the empirical covariation matrix converges in distribution almost surely. Our proof relies on method of moments and applications of graph theory.

math.PR↗
arXiv2014

Testing the maximal rank of the volatility process for continuous diffusions observed with noise

In this paper, we present a test for the maximal rank of the volatility process in continuous diffusion models observed with noise. Such models are typically applied in mathematical finance, where latent price processes are corrupted by microstructure noise at ultra high frequencies. Using high frequency observations we construct a test statistic for the maximal rank of the time varying stochastic volatility process. Our methodology is based upon a combination of a matrix perturbation approach and pre-averaging. We will show the asymptotic mixed normality of the test statistic and obtain a consistent testing procedure.

math.ST↗
arXiv2014

Limit theorems for nondegenerate U-statistics of continuous semimartingales

This paper presents the asymptotic theory for nondegenerate $U$-statistics of high frequency observations of continuous Itô semimartingales. We prove uniform convergence in probability and show a functional stable central limit theorem for the standardized version of the $U$-statistic. The limiting process in the central limit theorem turns out to be conditionally Gaussian with mean zero. Finally, we indicate potential statistical applications of our probabilistic results.

math.PR↗
arXiv2014

Ambit fields: survey and new challenges

In this paper we present a survey on recent developments in the study of ambit fields and point out some open problems. Ambit fields is a class of spatio-temporal stochastic processes, which by its general structure constitutes a flexible model for dynamical structures in time and/or in space. We will review their basic probabilistic properties, main stochastic integration concepts and recent limit theory for high frequency statistics of ambit fields.

math.PR↗
arXiv2014

On non-standard limits of Brownian semi-stationary processes

In this paper we present some new asymptotic results for high frequency statistics of Brownian semi-stationary processes. More precisely, we will show that singularities in the weight function, which is one of the ingredients of a BSS process, may lead to non-standard limits of the realised quadratic variation. In this case the limiting process is a convex combination of shifted integrals of the intermittency function. Furthermore, we will demonstrate the corresponding stable central limit theorem. Finally, we apply the probabilistic theory to study the asymptotic properties of the realised ratio statistics, which estimates the smoothness parameter of a BSS process.

math.PR↗
arXiv2014

High-frequency asymptotics for path-dependent functionals of Ito semimartingales

The estimation of local characteristics of Ito semimartingales has received a great deal of attention in both academia and industry over the past decades. In various papers limit theorems were derived for functionals of increments and ranges in the infill asymptotics setting. In this paper we establish the asymptotic theory for a wide class of statistics that are built from the incremental process of an Ito semimartingale. More specifically, we will show the law of large numbers and the associated stable central limit theorem for the path dependent functionals in the continuous and discontinuous framework. Some examples from economics and physics demonstrate the potential applicability of our theoretical results in practice.

math.ST↗
arXiv2014

Asymptotics of weighted random sums

In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We show that these sums converge in law to the integral of the weight process with respect to the Brownian motion when the distance between observations goes to zero. The result is obtained with the help of fractional calculus showing the power of this technique. This study, though interesting by itself, is motivated by an error found in the proof of Theorem 4 in Corcuera, J.M. Nualart, D., Woerner, J. H. C. (2006). Power variation of some integral fractional processes, Bernoulli 12(4) 713-735.

math.PR↗
arXiv2013

Edgeworth expansion for functionals of continuous diffusion processes

This paper presents new results on the Edgeworth expansion for high frequency functionals of continuous diffusion processes. We derive asymptotic expansions for weighted functionals of the Brownian motion and apply them to provide the second order Edgeworth expansion for power variation of diffusion processes. Our methodology relies on martingale embedding, Malliavin calculus and stable central limit theorems for semimartingales. Finally, we demonstrate the density expansion for studentized statistics of power variations.

math.PR↗
arXiv2012

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.

math.ST↗