Asymptotic distributions for the Ornstein-Uhlenbeck process

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.

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On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mathematics ◽

10.3390/math8112031 ◽

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).

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On the solution of Kac-type partial differential equations

Journal of Applied Probability ◽

10.1017/s0021900200107144 ◽

1994 ◽

Vol 31 (A) ◽

pp. 311-324

Author(s):

Mátyás Arató

Keyword(s):

Stochastic Process ◽

Cauchy Problem ◽

Brownian Motion ◽

Partial Differential Equations ◽

Differential Equations ◽

Explicit Form ◽

Uhlenbeck Process ◽

Ornstein Uhlenbeck Process ◽

The Cauchy Problem ◽

Constant Coefficients

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

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Asymptotic distributions of pontograms

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066470 ◽

1987 ◽

Vol 101 (1) ◽

pp. 131-139 ◽

Cited By ~ 10

Author(s):

Miklós Csörgő ◽

Lajos Horváth

Keyword(s):

Poisson Process ◽

Covariance Function ◽

Weighted Approximation ◽

Asymptotic Distributions ◽

Uhlenbeck Process ◽

Intensity Parameter ◽

Ornstein Uhlenbeck Process ◽

Image Position

Let{N(x), x ≥ 0} be a Poisson process with intensity parameter λ > 0, and introduceWhen looking for the changepoint in the Land's End data, Kendall and Kendall [7] proved for all 0 < ε1 < 1 − ε2 < 1 thatwhere {V(s), − ∞ < s < ∞} is an Ornstein–Uhlenbeck process with covariance function exp (−|t − s|). D. G. Kendall has posed the problem of replacing ε1 and ε2 by zero or by sequences εi(n) → 0 (n → ∞) (i = 1, 2), in (1·1). In this paper we study the latter problem and also its L2 version. The proofs will be based on the following weighted approximation of Zn.

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First-passage and first-exit times of a Bessel-like stochastic process

Physical Review E ◽

10.1103/physreve.83.051115 ◽

2011 ◽

Vol 83 (5) ◽

Cited By ~ 23

Author(s):

Edgar Martin ◽

Ulrich Behn ◽

Guido Germano

Keyword(s):

Stochastic Process ◽

Exit Times ◽

First Passage ◽

First Exit Times

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On the solution of Kac-type partial differential equations

Journal of Applied Probability ◽

10.2307/3214964 ◽

1994 ◽

Vol 31 (A) ◽

pp. 311-324

Author(s):

Mátyás Arató

Keyword(s):

Stochastic Process ◽

Cauchy Problem ◽

Brownian Motion ◽

Partial Differential Equations ◽

Differential Equations ◽

Explicit Form ◽

Uhlenbeck Process ◽

Ornstein Uhlenbeck Process ◽

The Cauchy Problem ◽

Constant Coefficients

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

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Hypothesis Testing in a Fractional Ornstein-Uhlenbeck Model

International Journal of Stochastic Analysis ◽

10.1155/2012/268568 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23 ◽

Cited By ~ 8

Author(s):

Michael Moers

Keyword(s):

Regression Analysis ◽

Unit Root ◽

Asymptotic Distributions ◽

Interesting Problem ◽

Finite Sample ◽

Asymptotic Power ◽

Uhlenbeck Process ◽

Sample Distribution ◽

Ornstein Uhlenbeck Process ◽

Drift Parameter

Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein's bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power.

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Invariant Solutions of Black–Scholes Equation with Ornstein–Uhlenbeck Process

Symmetry ◽

10.3390/sym13050847 ◽

2021 ◽

Vol 13 (5) ◽

pp. 847

Author(s):

Maba Boniface Matadi ◽

Phumlani Lawrence Zondi

Keyword(s):

Stochastic Process ◽

Option Pricing ◽

Invariant Solution ◽

Point Of View ◽

Theoretic Approach ◽

Invariant Solutions ◽

Uhlenbeck Process ◽

Independent Variables ◽

Black Scholes ◽

Ornstein Uhlenbeck Process

This paper analyses the model of Black–Scholes option pricing from the point of view of the group theoretic approach. The study identified new independent variables that lead to the transformation of the Black–Scholes equation. Furthermore, corresponding determining equations were constructed and new symmetries were found. As a result, the findings of the study demonstrate of the integrability of the model to present an invariant solution for the Ornstein–Uhlenbeck stochastic process.

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