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.

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.

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].

A computational approach to first-passage-time problems for Gauss–Markov processes

2001 ◽  
Vol 33 (2) ◽  
pp. 453-482 ◽  
Author(s):  
E. Di Nardo ◽  
A. G. Nobile ◽  
E. Pirozzi ◽  
L. M. Ricciardi
Keyword(s):  
Markov Processes ◽  
Brownian Bridge ◽  
First Passage Time ◽  
Accurate Method ◽  
Passage Time ◽  
Computational Approach ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
First Passage Time Problems

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

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 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.

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.

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.

scholarly journals A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 279
Author(s):  
Enrica Pirozzi
Keyword(s):  
Probability Density ◽  
Markov Processes ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
First Passage ◽  
Transition Probability Density ◽  
Symmetry Properties ◽  
Time Density

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

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