Construction of the Brownian motion and the Ornstein-Uhlenbeck process in a Riemannian manifold on basis of the Gangolli-Mc.Kean injection scheme

1978 ◽  
Vol 44 (1) ◽  
pp. 71-87 ◽  
Author(s):  
E. J�rgensen
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Uhlenbeck Process ◽  
Injection Scheme ◽  
Ornstein Uhlenbeck Process

scholarly journals On conditional Ornstein–Uhlenbeck processes

1984 ◽  
Vol 16 (04) ◽  
pp. 920-922
Author(s):  
P. Salminen
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Bessel Process ◽  
Uhlenbeck Process ◽  
Brownian Motions ◽  
Ornstein Uhlenbeck Process ◽  
The Law ◽  
Similar Description

It is well known that the law of a Brownian motion started from x > 0 and conditioned never to hit 0 is identical with the law of a three-dimensional Bessel process started from x. Here we show that a similar description is valid for all linear Ornstein–Uhlenbeck Brownian motions. Further, using the same techniques, it is seen that we may construct a non-stationary Ornstein–Uhlenbeck process from a stationary one.

scholarly journals On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2031
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Dynamical System ◽  
Stochastic Process ◽  
Brownian Motion ◽  
Markov Process ◽  
Markov Processes ◽  
Numerical Approximation ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Ornstein Uhlenbeck Process ◽  
Application Fields

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

Admission Control for Multidimensional Workload input with Heavy Tails and Fractional Ornstein-Uhlenbeck Process

2015 ◽  
Vol 47 (2) ◽  
pp. 476-505
Author(s):  
Amarjit Budhiraja ◽  
Vladas Pipiras ◽  
Xiaoming Song
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Admission Control ◽  
Control Policy ◽  
Uhlenbeck Process ◽  
Control Policies ◽  
Ornstein Uhlenbeck Process ◽  
Admission Control Policy

The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

scholarly journals Berry–Esséen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes

2019 ◽  
Vol 20 (04) ◽  
pp. 2050023 ◽  
Author(s):  
Yong Chen ◽  
Nenghui Kuang ◽  
Ying Li
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Continuous Time ◽  
Index Formula ◽  
Hurst Index ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Time Observation ◽  
Statistical Estimations ◽  
Drift Parameter

For an Ornstein–Uhlenbeck process driven by fractional Brownian motion with Hurst index [Formula: see text], we show the Berry–Esséen bound of the least squares estimator of the drift parameter based on the continuous-time observation. We use an approach based on Malliavin calculus given by Kim and Park [Optimal Berry–Esséen bound for statistical estimations and its application to SPDE, J. Multivariate Anal. 155 (2017) 284–304].

Admission Control for Multidimensional Workload input with Heavy Tails and Fractional Ornstein-Uhlenbeck Process

2015 ◽  
Vol 47 (02) ◽  
pp. 476-505
Author(s):  
Amarjit Budhiraja ◽  
Vladas Pipiras ◽  
Xiaoming Song
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Admission Control ◽  
Control Policy ◽  
Uhlenbeck Process ◽  
Control Policies ◽  
Ornstein Uhlenbeck Process ◽  
Admission Control Policy

The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

Generalized fractional Lévy processes with fractional Brownian motion limit

2015 ◽  
Vol 47 (04) ◽  
pp. 1108-1131 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Muneya Matsui
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Lévy Processes ◽  
Sample Path ◽  
Levy Processes ◽  
Order Structure ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Volatility Models ◽  
Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

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