A note on the Volterra integral equation for the first-passage-time probability density

1995 ◽  
Vol 32 (3) ◽  
pp. 635-648 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
P. Román Román ◽  
F. Torres Ruiz
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Volterra Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Closed Form Solutions ◽  
Actual Use ◽  
Probability Densities

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

A note on the Volterra integral equation for the first-passage-time probability density

1995 ◽  
Vol 32 (03) ◽  
pp. 635-648 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
P. Román Román ◽  
F. Torres Ruiz
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Volterra Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Closed Form Solutions ◽  
Actual Use ◽  
Probability Densities

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

A new integral equation for the evaluation of first-passage-time probability densities

1987 ◽  
Vol 19 (04) ◽  
pp. 784-800 ◽  
Author(s):  
A. Buonocore ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Continuous Functions ◽  
Time Dependent ◽  
First Passage ◽  
One Dimensional ◽  
Probability Densities

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

A new integral equation for the evaluation of first-passage-time probability densities

1987 ◽  
Vol 19 (4) ◽  
pp. 784-800 ◽  
Author(s):  
A. Buonocore ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Continuous Functions ◽  
Time Dependent ◽  
First Passage ◽  
One Dimensional ◽  
Probability Densities

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

On an integral equation for first-passage-time probability densities

1984 ◽  
Vol 21 (02) ◽  
pp. 302-314 ◽  
Author(s):  
L. M. Ricciardi ◽  
L. Sacerdote ◽  
S. Sato
Keyword(s):  
Integral Equation ◽  
Diffusion Process ◽  
Continuous Time ◽  
First Passage Time ◽  
Passage Time ◽  
Time Function ◽  
Standard Wiener Process ◽  
First Passage ◽  
Probability Densities ◽  
Bounded Derivative

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

1989 ◽  
Vol 26 (4) ◽  
pp. 707-721 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Constructive Approach ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities ◽  
Reflecting Boundaries ◽  
Natural Boundaries

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

A note on the evaluation of first-passage-time probability densities

1983 ◽  
Vol 20 (01) ◽  
pp. 197-201 ◽  
Author(s):  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Wiener Process ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Probability Densities

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

On an integral equation for first-passage-time probability densities

1984 ◽  
Vol 21 (2) ◽  
pp. 302-314 ◽  
Author(s):  
L. M. Ricciardi ◽  
L. Sacerdote ◽  
S. Sato
Keyword(s):  
Integral Equation ◽  
Diffusion Process ◽  
Continuous Time ◽  
First Passage Time ◽  
Passage Time ◽  
Time Function ◽  
Standard Wiener Process ◽  
First Passage ◽  
Probability Densities ◽  
Bounded Derivative

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

On the evaluation of first-passage-time probability densities via non-singular integral equations

1989 ◽  
Vol 21 (1) ◽  
pp. 20-36 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Singular Integral Equations ◽  
Passage Time ◽  
Time Dependent ◽  
Computational Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

A note on the evaluation of first-passage-time probability densities

1983 ◽  
Vol 20 (1) ◽  
pp. 197-201 ◽  
Author(s):  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Wiener Process ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Probability Densities

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

On the evaluation of first-passage-time probability densities via non-singular integral equations

1989 ◽  
Vol 21 (01) ◽  
pp. 20-36 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Singular Integral Equations ◽  
Passage Time ◽  
Time Dependent ◽  
Computational Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

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