An absorption probability for the Ornstein-Uhlenbeck process

1975 ◽  
Vol 12 (3) ◽  
pp. 595-599 ◽  
Author(s):  
John P. Dirkse
Keyword(s):  
Exact Solution ◽  
Asymptotic Behavior ◽  
Closed Form ◽  
Asymptotic Expression ◽  
Uhlenbeck Process ◽  
Absorption Probability ◽  
Optional Stopping ◽  
Ornstein Uhlenbeck Process ◽  
Kolmogorov Smirnov ◽  
Smirnov Statistic

An asymptotic expression for an absorption probability for the Ornstein-Uhlenbeck process is presented along with an application of the result to a problem in optional stopping. The relation of this result to the asymptotic behavior of a weighted Kolmogorov-Smirnov statistic is also discussed. Sweet and Hardin (1970) derive an exact solution (not in closed form) for this same problem.

An absorption probability for the Ornstein-Uhlenbeck process

1975 ◽  
Vol 12 (03) ◽  
pp. 595-599
Author(s):  
John P. Dirkse
Keyword(s):  
Exact Solution ◽  
Asymptotic Behavior ◽  
Closed Form ◽  
Asymptotic Expression ◽  
Uhlenbeck Process ◽  
Absorption Probability ◽  
Optional Stopping ◽  
Ornstein Uhlenbeck Process ◽  
Kolmogorov Smirnov ◽  
Smirnov Statistic

An asymptotic expression for an absorption probability for the Ornstein-Uhlenbeck process is presented along with an application of the result to a problem in optional stopping. The relation of this result to the asymptotic behavior of a weighted Kolmogorov-Smirnov statistic is also discussed. Sweet and Hardin (1970) derive an exact solution (not in closed form) for this same problem.

scholarly journals Exact solution for the anisotropic Ornstein–Uhlenbeck process

2021 ◽  
pp. 126526
Author(s):  
Rita M.C. de Almeida ◽  
Guilherme S.Y. Giardini ◽  
Mendeli Vainstein ◽  
James A. Glazier ◽  
Gilberto L. Thomas
Keyword(s):  
Exact Solution ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process

First-passage-time density and moments of the ornstein-uhlenbeck process

1988 ◽  
Vol 25 (01) ◽  
pp. 43-57 ◽  
Author(s):  
Luigi M. Ricciardi ◽  
Shunsuke Sato
Keyword(s):  
Asymptotic Behavior ◽  
First Passage Time ◽  
Passage Time ◽  
Exponential Functions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Time Density ◽  
Single Exponential ◽  
Explicit Expressions

A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.

First-passage-time density and moments of the ornstein-uhlenbeck process

1988 ◽  
Vol 25 (1) ◽  
pp. 43-57 ◽  
Author(s):  
Luigi M. Ricciardi ◽  
Shunsuke Sato
Keyword(s):  
Asymptotic Behavior ◽  
First Passage Time ◽  
Passage Time ◽  
Exponential Functions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Time Density ◽  
Single Exponential ◽  
Explicit Expressions

A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.

scholarly journals On a Generalization of One-Dimensional Kinetics

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1264
Author(s):  
Vladimir V. Uchaikin ◽  
Renat T. Sibatov ◽  
Dmitry N. Bezbatko
Keyword(s):  
Exact Solution ◽  
Asymptotic Behavior ◽  
Constant Velocity ◽  
Telegraph Equation ◽  
Material Derivative ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Special Cases ◽  
Fractional Telegraph Equation ◽  
Asymmetric Case

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

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