We propose a generalization of the Ornstein-Uhlenbeck process in 1 + 1 dimensions which is the product of a temporal Ornstein-Uhlenbeck process with a spatial one and has exponentially decaying autocorrelation. The generalized Langevin equation of the process, the corresponding Fokker-Planck equation, and a discrete integral algorithm for numerical simulation is given. The process is an alterna...

We combine earlier investigations of linear systems with Lévy fluc-We give a complete construction of the Ornstein-Uhlenbeck-Cauchy process as a fully computable model of an anomalous transport and a paradigm example of Doob's stable noise-supported Ornstein-Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators ...

Abstract: We consider an Ornstein-Uhlenbeck process with values in Rn driven by a Lévy process (Zt) taking values in R d with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the Lévy measure of (Zt), we prove that the law of the Ornstein-Uhlenbeck process at any time t > 0 has a density ...

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical non-steepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation...

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein-Uhlenbeck (O-U) process to the first hitting time distribution of an O...

Abstract: We consider an Ornstein-Uhlenbeck process with values in Rn driven by a Lévy process (Zt) taking values in R d with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Lévy measure of (Zt), we prove that the law of the Ornstein-Uhlenbeck process at any time t > 0 has a...

De Haan and Karandikar [12] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes (Vt)t≥0 which are basically characterized by the fact that for each h > 0 the equidistantly sampled process (Vnh)n∈N0 satisfies the random recurrence equation Vnh = A(n−1)h,nhV(n−1)h + B(n−1)h,nh, n ∈ N, where (A(n−1)h,nh, B(n−1)h,nh)n∈N is an i.i.d. sequence with positive A0,h for each ...

Abstract We prove asymptotically efficient inference results concerning an Ornstein–Uhlenbeck regression model driven by a non-Gaussian stable Lévy process, where the output process is observed at high frequency over fixed period. The local asymptotics of non-ergodic type for likelihood function presented, followed way to construct estimator through suboptimal, yet very simple preliminary estim...

In this paper we consider an Ornstein-Uhlenbeck (ou) process (M(t))t>0 whose parameters are determined by an external Markov process (X(t))t>0 on a nite state space {1, . . . , d}; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (or: mmou). We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a sys...