On the two-boundary first-crossing-time problem for diffusion processes

1990 ◽  
Vol 27 (1) ◽  
pp. 102-114 ◽  
Author(s):  
A. Buonocore ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equations ◽  
Diffusion Processes ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Time Problem ◽  
Crossing Time ◽  
Single Integral ◽  
First Crossing Time

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

On the two-boundary first-crossing-time problem for diffusion processes

1990 ◽  
Vol 27 (01) ◽  
pp. 102-114 ◽  
Author(s):  
A. Buonocore ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equations ◽  
Diffusion Processes ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Time Problem ◽  
Crossing Time ◽  
Single Integral ◽  
First Crossing Time

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

1995 ◽  
Vol 32 (2) ◽  
pp. 316-336 ◽  
Author(s):  
A. G. Di Crescenzo ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Processes ◽  
Linear Process ◽  
Two Dimensional ◽  
Constructive Approach ◽  
One Dimensional ◽  
Absorbing Boundaries ◽  
Dimensional Diffusion ◽  
Crossing Time ◽  
Probability Densities ◽  
First Crossing Time

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

1995 ◽  
Vol 32 (02) ◽  
pp. 316-336 ◽  
Author(s):  
A. G. Di Crescenzo ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Processes ◽  
Linear Process ◽  
Two Dimensional ◽  
Constructive Approach ◽  
One Dimensional ◽  
Absorbing Boundaries ◽  
Dimensional Diffusion ◽  
Crossing Time ◽  
Probability Densities ◽  
First Crossing Time

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

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].

scholarly journals Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

2014 ◽  
Vol 46 (01) ◽  
pp. 186-202 ◽  
Author(s):  
Laura Sacerdote ◽  
Ottavia Telve ◽  
Cristina Zucca
Keyword(s):  
Diffusion Process ◽  
Joint Distribution ◽  
Continuous Functions ◽  
Time Dependent ◽  
Hitting Times ◽  
Convergence Properties ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
First Hitting Times

Consider a one-dimensional diffusion process on the diffusion interval I originated in x 0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t 0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t 0. We study the joint distribution of the two random variables T a and T b , the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of T a and T b in terms of ℙ(T a < t, T a < T b ) and ℙ(T b < t, T a > T b ), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

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].

scholarly journals Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

2014 ◽  
Vol 46 (1) ◽  
pp. 186-202 ◽  
Author(s):  
Laura Sacerdote ◽  
Ottavia Telve ◽  
Cristina Zucca
Keyword(s):  
Diffusion Process ◽  
Joint Distribution ◽  
Continuous Functions ◽  
Time Dependent ◽  
Hitting Times ◽  
Convergence Properties ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
First Hitting Times

Consider a one-dimensional diffusion process on the diffusion interval I originated in x0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t0. We study the joint distribution of the two random variables Ta and Tb, the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta and Tb in terms of ℙ(Ta < t, Ta < Tb) and ℙ(Tb < t, Ta > Tb), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

The Level-Crossing Problem: First-Passage, Escape and Extremes

2014 ◽  
Vol 13 (04) ◽  
pp. 1430001 ◽  
Author(s):  
Jaume Masoliver
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Level Crossing ◽  
First Passage ◽  
One Dimensional ◽  
Absolute Value ◽  
Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

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