On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary

1987 ◽  
Vol 24 (02) ◽  
pp. 355-369
Author(s):  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Integral Type ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Probability Current ◽  
Ornstein Uhlenbeck Process ◽  
Probability Densities ◽  
Free Transition ◽  
Absorption Condition

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary

1987 ◽  
Vol 24 (2) ◽  
pp. 355-369 ◽  
Author(s):  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Integral Type ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Probability Current ◽  
Ornstein Uhlenbeck Process ◽  
Probability Densities ◽  
Free Transition ◽  
Absorption Condition

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

1985 ◽  
Vol 22 (02) ◽  
pp. 360-369 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Asymptotic Behaviour ◽  
Excellent Agreement ◽  
First Passage Time ◽  
Passage Time ◽  
Computational Results ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Exponential Behaviour

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

scholarly journals A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

2011 ◽  
Vol 48 (02) ◽  
pp. 420-434 ◽  
Author(s):  
Peter J. Thomas
Keyword(s):  
Lower Bound ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Condition ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Fire Model ◽  
Integrate And Fire ◽  
Ornstein Uhlenbeck Process ◽  
Time Density

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck processX(t) obeying dX= -βXdt+ σdWto reach a fixed threshold θ from a suprathreshold initial conditionx0> θ > 0 has a lower bound of the form ρ(t) >kexp[-pe6βt] for positive constantskandpfor timestexceeding some positive valueu. We obtain explicit expressions fork,p, anduin terms of β, σ,x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

Evaluation of the first-passage time probability to a square root boundary for the Wiener process

1977 ◽  
Vol 14 (4) ◽  
pp. 850-856 ◽  
Author(s):  
Shunsuke Sato
Keyword(s):  
Wiener Process ◽  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
Square Root ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Asymptotic Evaluation

This paper gives an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary. The result is applied to estimate the moments of the first-passage time distribution of the Ornstein–Uhlenbeck process to a constant boundary.

A Recursion Formula for the Moments of the First Passage Time of the Ornstein-Uhlenbeck Process

2015 ◽  
Vol 52 (02) ◽  
pp. 595-601
Author(s):  
Dirk Veestraeten
Keyword(s):  
Lower Order ◽  
First Passage Time ◽  
Recursion Formula ◽  
Passage Time ◽  
Higher Order ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Higher Order Moments ◽  
Ornstein Uhlenbeck Process

In this paper we use the Siegert formula to derive alternative expressions for the moments of the first passage time of the Ornstein-Uhlenbeck process through a constant threshold. The expression for the nth moment is recursively linked to the lower-order moments and consists of only n terms. These compact expressions can substantially facilitate (numerical) applications also for higher-order moments.

Evaluation of the first-passage time probability to a square root boundary for the Wiener process

1977 ◽  
Vol 14 (04) ◽  
pp. 850-856 ◽  
Author(s):  
Shunsuke Sato
Keyword(s):  
Wiener Process ◽  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
Square Root ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Asymptotic Evaluation

This paper gives an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary. The result is applied to estimate the moments of the first-passage time distribution of the Ornstein–Uhlenbeck process to a constant boundary.

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