Estimating Drift Parameters in a Fractional Ornstein Uhlenbeck Process with Periodic Mean

We construct a least squares estimator for the drift parameters of a fractional Ornstein Uhlenbeck process with periodic mean function and long range dependence. For this estimator we prove consistency and asymptotic normality. In contrast to the classical fractional Ornstein Uhlenbeck process without periodic mean function the rate of convergence is slower depending on the Hurst parameter $H$, namely $n^{1-H}$

A unifying approach to fractional Lévy processes

Starting from the moving average representation of fractional Brownian motion fractional Lévy processes have been constructed by keeping the same moving average kernel and replacing the Brownian motion by a pure jump Lévy process with finite second moments. Another way was to replace the Brownian motion by an alpha-stable Lévy process and the exponent in the kernel by H-1/alpha. We now provide a unifying approach taking kernels of the form a((t-s)_+^gamma - (-s)_+^gamma) + b((t-s)_-^gamma - (-s)_-^gamma), where gamma can be chosen according to the existing moments and the Blumenthal-Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g. regularity of the sample paths and the semimartingale property. MSC 2010: 60G22, 60E0

Functional central limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $\beta$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers. Here we provide associated functional central limit theorems which are locally uniform in $t$. The Gaussian limiting processes admit explicit representations in terms of matrix exponentials and the solutions of the associated deterministic dynamical systems.Comment: This is an abridged version of the previous paper without the ODE parts. The ODE part in an extended form can be found in Arxiv:1910.0788

Well-balanced Lévy driven Ornstein-Uhlenbeck processes

In this paper we introduce the well-balanced Lévy driven Ornstein-Uhlenbeck process as a moving average process of the form X_t=integral(exp(-lambda*|t-u|)dL_u). In contrast to Lévy driven Ornstein-Uhlenbeck processes the well-balanced form possesses continuous sample paths and an autocorrelation function which is decreasing more slowly. Furthermore, depending on the size of lambda it allows both for positive and negative correlation of increments. As Ornstein-Uhlenbeck processes X_t is a stationary process starting at X_0=integral(exp(-lambda*u)dL_u). However, by taking a difference kernel we can construct a process with stationary increments starting at zero, which possesses the same correlation structure