A note on first-passage time and some related problems

1985 ◽  
Vol 22 (2) ◽  
pp. 346-359 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

A note on first-passage time and some related problems

1985 ◽  
Vol 22 (02) ◽  
pp. 346-359 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

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.

Construction of first-passage-time densities for a diffusion process which is not necessarily time-homogeneous

1991 ◽  
Vol 28 (4) ◽  
pp. 903-909 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
A. Juan Gonzalez ◽  
P. Román Román
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
First Passage Time ◽  
Passage Time ◽  
Probability Density Functions ◽  
New Method ◽  
Kolmogorov Equation ◽  
Density Functions ◽  
First Passage ◽  
Time Transformations

In Giorno et al. (1988) a new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new time-homogeneous diffusion process in terms of a preassigned transition p.d.f. without making use of the classical space-time transformations of the Kolmogorov equation (Ricciardi (1976)).In the present paper we give an extension of this result to the case of a diffusion process X(t) which is not necessarily time-homogeneous, and a few examples are presented.

scholarly journals A Note on First Passage Functionals for Lévy Processes with Jumps of Rational Laplace Transforms

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Djilali Ait-Aoudia
Keyword(s):  
Boundary Value Problem ◽  
Laplace Transform ◽  
Explicit Formula ◽  
First Passage Time ◽  
Boundary Value ◽  
Jump Process ◽  
Passage Time ◽  
Laplace Transforms ◽  
First Passage ◽  
Exit Problem

This paper investigates the two-sided first exit problem for a jump process having jumps with rational Laplace transform. The corresponding boundary value problem is solved to obtain an explicit formula for the first passage functional. Also, we derive the distribution of the first passage time to two-sided barriers and the value at the first passage time.

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.

Construction of first-passage-time densities for a diffusion process which is not necessarily time-homogeneous

1991 ◽  
Vol 28 (04) ◽  
pp. 903-909 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
A. Juan Gonzalez ◽  
P. Román Román
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
First Passage Time ◽  
Passage Time ◽  
Probability Density Functions ◽  
New Method ◽  
Kolmogorov Equation ◽  
Density Functions ◽  
First Passage ◽  
Time Transformations

In Giorno et al. (1988) a new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new time-homogeneous diffusion process in terms of a preassigned transition p.d.f. without making use of the classical space-time transformations of the Kolmogorov equation (Ricciardi (1976)). In the present paper we give an extension of this result to the case of a diffusion process X(t) which is not necessarily time-homogeneous, and a few examples are presented.

scholarly journals Critical scaling for the SIS stochastic epidemic

2006 ◽  
Vol 43 (3) ◽  
pp. 892-898 ◽  
Author(s):  
R. G. Dolgoarshinnykh ◽  
Steven P. Lalley
Keyword(s):  
Diffusion Process ◽  
First Passage Time ◽  
Scaling Law ◽  
Passage Time ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Sir Epidemic ◽  
Stochastic Epidemic ◽  
Ornstein Uhlenbeck Process ◽  
Sis Epidemic

We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of infected and susceptible individuals, then when the time and the number currently infected are both scaled by , the resulting process converges, as N → ∞, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Löf (1998) and Aldous (1997) for the simple SIR epidemic.

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.

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