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.

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.

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 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.

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].

The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type

2020 ◽  
Vol 28 (1) ◽  
pp. 43-52
Author(s):  
Durdimurod Kalandarovich Durdiev ◽  
Zhanna Dmitrievna Totieva
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equations ◽  
Stability Estimate ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Weighted Norms ◽  
The Stability ◽  
The One ◽  
Global Unique Solvability

AbstractThe integro-differential system of viscoelasticity equations with a source of explosive type is considered. It is assumed that the coefficients of the equations depend only on one spatial variable. The problem of determining the kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to one inverse problem for scalar hyperbolic equations. This inverse problem is replaced by an equivalent system of integral equations for unknown functions. The principle of constricted mapping in the space of continuous functions with weighted norms to the latter is applied. The theorem of global unique solvability is proved and the stability estimate of solution to the inverse problem is obtained.

scholarly journals LQG Homing in a Finite Time Interval

2011 ◽  
Vol 2011 ◽  
pp. 1-3 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Optimal Control ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Fixed Constant

LetX(t)be a controlled one-dimensional diffusion process having constant infinitesimal variance. We consider the problem of optimally controllingX(t)until timeT(x)=min{T1(x),t1}, whereT1(x)is the first-passage time of the process to a given boundary andt1is a fixed constant. The optimal control is obtained explicitly in the particular case whenX(t)is a controlled Wiener process.

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