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.

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.

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 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.

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

1989 ◽  
Vol 26 (04) ◽  
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.

First-passage-time densities for time-non-homogeneous diffusion processes

1997 ◽  
Vol 34 (3) ◽  
pp. 623-631 ◽  
Author(s):  
R. Gutiérrez ◽  
L. M. Ricciardi ◽  
P. Román ◽  
F. Torres
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Continuous Functions ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

1990 ◽  
Vol 22 (4) ◽  
pp. 883-914 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
Asymptotic Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Periodic Boundaries

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

scholarly journals On first-passage-time and transition densities for strongly symmetric diffusion processes

1997 ◽  
Vol 145 ◽  
pp. 143-161 ◽  
Author(s):  
A. Di Crescenzo ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Process ◽  
Closed Form ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
Closed Form Expression ◽  
First Passage ◽  
One Dimensional ◽  
Transition Functions

One dimensional diffusion processes have been increasingly invoked to model a variety of biological, physical and engineering systems subject to random fluctuations (cf., for instance, Blake, I. F. and Lindsey, W. C. [2], Abrahams, J. [1], Giorno, V. et al [10] and references therein). However, usually the knowledge of the ‘free’ transition probability density function (pdf) is not sufficient; one is thus led to the more complicated task of determining transition functions in the presence of preassigned absorbing boundaries, or first-passage-time densities for time-dependent boundaries (see, for instance, Daniels, H. E. [6], [7], Giorno, V. et al. [10]). Such densities are known analytically only in some special instances so that numerical methods have to be implemented in general (cf., for instance, Buono-core, A. et al [3], [4], Giorno, V. et al [11]). The analytical approach becomes particularly effective when the diffusion process exhibits some special features, such as the symmetry of its transition pdf. For instance, in [10] special symmetry conditions on the transition pdf of one-dimensional time-homogeneous diffusion process with natural boundaries are investigated to derive closed form results concerning the transition pdf’s and the first-passage-time pdf for particular time-dependent boundaries. On the other hand, by using the method of images, in [6] Daniels has obtained a closed form expression for the transition pdf of the standard Wiener process in the presence of a particular time-dependent absorbing boundary. It is interesting to remark that such density cannot be obtained via the methods described in [10], even though the considered process exhibits the kind of symmetry discussed therein.

First-passage-time densities for time-non-homogeneous diffusion processes

1997 ◽  
Vol 34 (03) ◽  
pp. 623-631 ◽  
Author(s):  
R. Gutiérrez ◽  
L. M. Ricciardi ◽  
P. Román ◽  
F. Torres
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Continuous Functions ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

1990 ◽  
Vol 22 (04) ◽  
pp. 883-914 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
Asymptotic Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Periodic Boundaries

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

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