We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any $L^2$ functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model.
We investigate the asymptotic properties of the minimum $L_1$-norm estimator of the drift parameter for fractional Ornstein-Uhlenbeck type process driven by a general Gaussian process.
In this article, we introduce a non Gaussian long memory process constructed by the aggregation of independent copies of a fractional Lévy Ornstein-Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown.
Higher-order Derivative Local Time for Fractional Ornstein-Uhlenbeck Processes
本文考虑了两个d维分数Ornstein-Uhlenbeck过程$X^{H_1}_t$和$\widetilde{X}^{H_2}_s$分别具有Hurst参数$H_1$和$H_2$的局部时$ \widehatα^{(k)}(x,t)$的$k$阶导数的存在性.此外，利用Guo et al.（2017）和Lou et al.（2017）中的一些技巧，获得了局部时的分数阶Ornstein-Uhlenbeck过程$X^{H}_t$的H $\hat{o}$ lder正则性条件$\tildeα^{(k)}(x,t)$。
In this paper, we study Ornstein-Uhlenbeck processes with Markov modulation, whose parameters depend on an external underlying two-state Markov process. Conditional mean and variance of such processes under given modulation are investigated from the point of view of the first passage probabilities and invariant measures. It is also studied the limiting behaviour under scaling conditions similar to Kac's scaling.
In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein-Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein-Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorohod $J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H \to 1/…
In this paper we study some properties of the generalized Fokker-Planck equation induced by the time-changed fractional Ornstein-Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.
In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein-Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter $H>1/2$.
Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval $I$ such that the process starting from the internal point of $I$ always remains within $I$. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the t…
