On the First Passage Time of Diffusion Processes Through Time-Dependent Boundaries

1971 ◽  
Vol 16 (3) ◽  
pp. 541-545
Author(s):  
V. A. Lebedev
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
First Passage

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 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 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.

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 (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 first passage times of sticky reflecting diffusion processes with double exponential jumps

2020 ◽  
Vol 57 (1) ◽  
pp. 221-236 ◽  
Author(s):  
Shiyu Song ◽  
Yongjin Wang
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Explicit Solutions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Double Exponential ◽  
Upper Level ◽  
First Passage Problem ◽  
Passage Times

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

scholarly journals Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 555 ◽  
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Time Dependent ◽  
Transition Probability Density ◽  
Special Boundaries ◽  
Special Cases ◽  
Regulation Functions

We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.

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