arXiv subjects

Mark Podolskij

Publications and source records attributed to Mark Podolskij.

At least 37 records · Page 2Linked to original sources

arXiv2024

Optimal estimation of local time and occupation time measure for an α-stable Levy process

We present a novel theoretical result on estimation of local time and occupation time measure of an α-stable Lévy process with α in (1, 2). Our approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an L^2-optimal statistic by construction. We prove the corresponding stable central limit theorems and discuss a statistical application. In particular, this work extends the results of [Ivanovs and i Podolskij (2021)], which investigated the case of the Brownian motion.

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arXiv2023

Limit theorems for general functionals of Brownian local times

In this paper, we present the asymptotic theory for integrated functions of increments of Brownian local times in space. Specifically, we determine their first-order limit, along with the asymptotic distribution of the fluctuations. Our key result establishes that a standardized version of our statistic converges stably in law towards a mixed normal distribution. Our contribution builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li, M.B. Markus, D. Nualart and J. Rosen \cite{C17,CLMR10,HN09,HN10,MR08,R11,R11b}, which delved into special cases of the considered problem, such as quadratic, cubic and polynomial cases. We establish the limit theorem for general functions that satisfy mild smoothness and growth conditions. This extends the scope beyond the polynomial cases studied in previous works, providing a more comprehensive understanding of the asymptotic properties of the considered functionals.

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arXiv2023

Parameter estimation of discretely observed interacting particle systems

In this paper, we consider the problem of joint parameter estimation for drift and diffusion coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval $[0, T]$, we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($Δ_n$) and the number of particles ($N$) satisfy $Δ_n \rightarrow 0$ and $N \rightarrow \infty$, and asymptotically normal when additionally the condition $Δ_n N \rightarrow 0$ holds.

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arXiv2023

On Lasso estimator for the drift function in diffusion models

In this paper we study the properties of the Lasso estimator of the drift component in the diffusion setting. More specifically, we consider a multivariate parametric diffusion model $X$ observed continuously over the interval $[0,T]$ and investigate drift estimation under sparsity constraints. We allow the dimensions of the model and the parameter space to be large. We obtain an oracle inequality for the Lasso estimator and derive an error bound for the $L^2$-distance using concentration inequalities for linear functionals of diffusion processes. The probabilistic part is based upon elements of empirical processes theory and, in particular, on the chaining method.

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arXiv2023

Asymptotic theory for quadratic variation of harmonizable fractional stable processes

In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional $\al$-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a Lévy-driven Rosenblatt random variable when the Hurst parameter satisfies $H\in (1/2,1)$ and $\al(1-H)<1/2$. This result complements the asymptotic theory for fractional stable processes investigated in e.g. \cite{BHP19,BLP17,BP17,BPT20,LP18,MOP20}.

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arXiv2022

Estimation of mixed fractional stable processes using high-frequency data

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. Applying an estimating equations approach, we construct estimators for the whole set of parameters and derive their asymptotic normality in a high-frequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, i.e. for the estimation of the successive Blumenthal-Getoor indices, and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. Our asymptotic theory is based on new limit theorems for multiscale moving average processes.

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arXiv2022

Power variations for fractional type infinitely divisible random fields

This paper presents new limit theorems for power variation of fractional type symmetric infinitely divisible random fields. More specifically, the random field $X = (X(\boldsymbol{t}))_{\boldsymbol{t} \in [0,1]^d}$ is defined as an integral of a kernel function $g$ with respect to a symmetric infinitely divisible random measure $L$ and is observed on a grid with mesh size $n^{-1}$. As $n \to \infty$, the first order limits are obtained for power variation statistics constructed from rectangular increments of $X$. The present work is mostly related to Basse-O'Connor, Lachièze-Rey, Podolskij (2017), Basse-O'Connor, Heinrich, Podolskij (2019), who studied a similar problem in the case $d=1$. We will see, however, that the asymptotic theory in the random field setting is much richer compared to Basse-O'Connor, Lachièze-Rey, Podolskij (2017), Basse-O'Connor, Heinrich, Podolskij (2019) as it contains new limits, which depend on the precise structure of the kernel $g$. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional $β$-stable sheet, and the moving average fractional $β$-stable field, and discuss potential consequences for statistical inference.

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arXiv2021

Semiparametric estimation of McKean-Vlasov SDEs

In this paper we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.

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arXiv2021

On estimation of quadratic variation for multivariate pure jump semimartingales

In this paper we present the asymptotic analysis of the realised quadratic variation for multivariate symmetric $β$-stable Lévy processes, $β\in (0,2)$, and certain pure jump semimartingales. The main focus is on derivation of functional limit theorems for the realised quadratic variation and its spectrum. We will show that the limiting process is a matrix-valued $β$-stable Lévy process when the original process is symmetric $β$-stable, while the limit is conditionally $β$-stable in case of integrals with respect to symmetric $β$-stable motions. These asymptotic results are mostly related to the work [5], which investigates the univariate version of the problem. Furthermore, we will show the implications for estimation of eigenvalues and eigenvectors of the quadratic variation matrix, which is a useful result for the principle component analysis. Finally, we propose a consistent subsampling procedure in the Lévy setting to obtain confidence regions.

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arXiv2021

Multi-dimensional parameter estimation of heavy-tailed moving averages

In this paper we present a parametric estimation method for certain multi-parameter heavy-tailed Lévy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained in [3] via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to the papers [14, 15], which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multi-parametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

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arXiv2020

On Dantzig and Lasso estimators of the drift in a high dimensional Ornstein-Uhlenbeck model

In this paper we present new theoretical results for the Dantzig and Lasso estimators of the drift in a high dimensional Ornstein-Uhlenbeck model under sparsity constraints. Our focus is on oracle inequalities for both estimators and error bounds with respect to several norms. In the context of the Lasso estimator our paper is strongly related to [11], who investigated the same problem under row sparsity. We improve their rates and also prove the restricted eigenvalue property solely under ergodicity assumption on the model. Finally, we demonstrate a numerical analysis to uncover the finite sample performance of the Dantzig and Lasso estimators.

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arXiv2020

A Minimal Contrast Estimator for the Linear Fractional Stable Motion

In this paper we present an estimator for the three-dimensional parameter $(σ, α, H)$ of the linear fractional stable motion, where $H$ represents the self-similarity parameter, and $(σ, α)$ are the scaling and stability parameters of the driving symmetric Lévy process $L$. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the selfsimilarity parameter $H$. The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to [16, 18], in which parameter estimation for the linear fractional stable motion and related Lévy moving average processes has been studied.

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arXiv2020

Optimal estimation of some random quantities of a Lévy process

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time measure of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators. Furthermore, we discuss pre-estimation of the parameters of the underlying models, which is required for practical implementation of the proposed statistics.

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arXiv2019

The asymptotic error of chaos expansion approximations for stochastic differential equations

In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the $L^2$ approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.

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arXiv2019

A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages

In this paper we obtain Berry-Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $α$, and its tail-index, which is controlled by a parameter $β$. In fact, we obtain the classical $\sqrt{1/ n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $αβ> 3$ or $αβ> 4$ in the case of Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely on a new second-order Poincare inequality on the Poisson space, which we derive through a combination of Stein's method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].

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arXiv2018

Edgeworth expansion for Euler approximation of continuous diffusion processes

In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work \cite{Yoshida2013}, which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.

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arXiv2018

A limit theorem for a class of stationary increments Lévy moving average process with multiple singularities

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477--4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360--383] under the assumption that the kernel function potentially exhibits a singular behaviour at $0$. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653--677] that studied the same mathematical question for the class of Brownian semi-stationary models.

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arXiv2018

On limit theory for functionals of stationary increments Levy driven moving averages

In this paper we obtain new limit theorems for variational functionals of high frequency observations of stationary increments Lévy driven moving averages. We will see that the asymptotic behaviour of such functionals heavily depends on the kernel, the driving Lévy process and the properties of the functional under consideration. We show the "law of large numbers" for our class of statistics, which consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we also prove the associated weak limit theory, which again consists of three different cases. Our work is related to [9,10], who considered power variation functionals of stationary increments Lévy driven moving averages.

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