The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold-Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian-and Lévy-driven fractional Ornstein-Uhlenbeck processes are presented. A Fubini theorem for Lévy bases is established as an element in the derivations.1 2 E[(X t − X 0 ) 2 ] its complementary autocovariance function.Before discussing the general setting further, we recall some well-known cases. The stationary solution X to (2.1) when N t = µt + σB t (with B the Brownian motion) is the

In this paper, we present some new limit theorems for power variation of kth 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 novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k ≥ 1, the considered power p > 0, the BlumenthalGetoor index β ∈ [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 a 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 (k − α)β-stable totally right skewed random variable.

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.

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a continuous-time equivalent of the AR(∞) representation. Furthermore, we show how this representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.

In this paper we present some limit theorems for power variation of stationary increments Lévy driven moving averages in the setting of critical regimes. In [5] the authors derived first and second order asymptotic results for k-th order increments of stationary increments Lévy driven moving averages. The limit theory heavily depends on the interplay between the given order of the increments, the considered power, the Blumenthal-Getoor index of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0. In this work we will study the critical cases, which were not covered in the original work [5].