Moments of the first-passage time of a Wiener process with drift between two elastic barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

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First passage time distribution of a Wiener process with drift concerning two elastic barriers

Journal of Applied Probability ◽

10.1017/s0021900200103821 ◽

1996 ◽

Vol 33 (01) ◽

pp. 164-175 ◽

Cited By ~ 5

Author(s):

Marco Dominé

Keyword(s):

Explicit Expression ◽

Wiener Process ◽

Time Distribution ◽

First Passage Time ◽

Planck Equation ◽

Passage Time ◽

Special Choice ◽

First Passage ◽

Special Cases ◽

Reflecting Barriers

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

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First passage time distribution of a Wiener process with drift concerning two elastic barriers

Journal of Applied Probability ◽

10.2307/3215274 ◽

1996 ◽

Vol 33 (1) ◽

pp. 164-175 ◽

Cited By ~ 22

Author(s):

Marco Dominé

Keyword(s):

Explicit Expression ◽

Wiener Process ◽

Time Distribution ◽

First Passage Time ◽

Planck Equation ◽

Passage Time ◽

Special Choice ◽

First Passage ◽

Special Cases ◽

Reflecting Barriers

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

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The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

International Journal of Stochastic Analysis ◽

10.1155/2012/971212 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15 ◽

Cited By ~ 4

Author(s):

Chuancun Yin ◽

Huiqing Wang

Keyword(s):

Boundary Conditions ◽

Value Function ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

One Dimensional ◽

The Laplace Transform ◽

The Value Function ◽

Reflecting Barriers

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered.

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On the First-Passage Distribution for the Envelope of a Nonstationary Narrow-Band Stochastic Process

Journal of Applied Mechanics ◽

10.1115/1.3423390 ◽

1974 ◽

Vol 41 (3) ◽

pp. 793-797 ◽

Cited By ~ 31

Author(s):

W. C. Lennox ◽

D. A. Fraser

Keyword(s):

Stochastic Process ◽

Hypergeometric Functions ◽

Narrow Band ◽

First Passage Time ◽

Wide Band ◽

Passage Time ◽

First Passage ◽

Eigenvalues And Eigenfunctions ◽

One Dimensional ◽

Backward Equation

A narrow-band stochastic process is obtained by exciting a lightly damped linear oscillator by wide-band stationary noise. The equation describing the envelope of the process is replaced, in an asymptotic sense, by a one-dimensional Markov process and the modified Kolmogorov (backward) equation describing the first-passage distribution function is solved exactly using classical methods by reducing the problem to that of finding the related eigenvalues and eigenfunctions; in this case degenerate hypergeometric functions. If the exciting process is white noise, the analysis is exact. The method also yields reasonable approximations for the first-passage time of the actual narrow-band process for either a one-sided or a symmetric two-sided barrier.

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First Passage Problems for Asymmetric Wiener Processes

Journal of Applied Probability ◽

10.1239/jap/1143936251 ◽

2006 ◽

Vol 43 (1) ◽

pp. 175-184 ◽

Cited By ~ 7

Author(s):

Mario Lefebvre

Keyword(s):

Wiener Process ◽

Generating Function ◽

First Passage Time ◽

Moment Generating Function ◽

Passage Time ◽

Expected Value ◽

First Passage ◽

One Dimensional ◽

Wiener Processes ◽

The Moment

The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

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First Passage Problems for Asymmetric Wiener Processes

Journal of Applied Probability ◽

10.1017/s0021900200001443 ◽

2006 ◽

Vol 43 (01) ◽

pp. 175-184

Author(s):

Mario Lefebvre

Keyword(s):

Wiener Process ◽

Generating Function ◽

First Passage Time ◽

Moment Generating Function ◽

Passage Time ◽

Expected Value ◽

First Passage ◽

One Dimensional ◽

Wiener Processes ◽

The Moment

The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

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First-Passage Time in a Random Vibrational System

Journal of Applied Mechanics ◽

10.1115/1.3624977 ◽

1966 ◽

Vol 33 (1) ◽

pp. 187-191 ◽

Cited By ~ 18

Author(s):

A. H. Gray

Keyword(s):

First Passage Time ◽

Passage Time ◽

Time To Failure ◽

Mean Square ◽

First Passage ◽

One Dimensional ◽

First Order ◽

Vibrational System ◽

The Mean ◽

First Passage Problem

The first-passage problem in a random vibrational system is solved by means of approximations which convert the problem to a first-order one-dimensional Markov process. Laplace transforms are used to evaluate the mean square time to failure.

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