scholarly journals Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Noise Intensity ◽  
First Passage Time ◽  
Transition Density ◽  
Passage Time ◽  
Intensity Function ◽  
Absorbing Boundary Condition ◽  
First Passage ◽  
Absorbing Boundary ◽  
Time Density

AbstractA time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift $$\alpha (t)x+\beta (t)$$ α ( t ) x + β ( t ) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function $$\beta (t)$$ β ( t ) and the noise intensity function r(t) are connected via the relation $$\beta (t)=\xi \,r(t)$$ β ( t ) = ξ r ( t ) , with $$0\le \xi <1$$ 0 ≤ ξ < 1 . Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that $$\alpha (t)$$ α ( t ) , $$\beta (t)$$ β ( t ) or both of these functions exhibit some kind of periodicity.

scholarly journals Mean first-passage time of a tumor cell growth system with time delay and colored cross-correlated noises excitation

2017 ◽  
Vol 37 (2) ◽  
pp. 191-198 ◽  
Author(s):  
Shenghong Li ◽  
Yong Huang
Keyword(s):  
Time Delay ◽  
Noise Intensity ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
Tumor Cell Growth ◽  
First Passage ◽  
Growth System ◽  
The Mean ◽  
Correlated Noises

In this paper, the mean first-passage time of a delayed tumor cell growth system driven by colored cross-correlated noises is investigated. Based on the Novikov theorem and the method of probability density approximation, the stationary probability density function is obtained. Then applying the fastest descent method, the analytical expression of the mean first-passage time is derived. Finally, effects of different kinds of delays and noise parameters on the mean first-passage time are discussed thoroughly. The results show that the time delay included in the random force, additive noise intensity and multiplicative noise intensity play a positive role in the disappearance of tumor cells. However, the time delay included in the determined force and the correlation time lead to the increase of tumor cells.

scholarly journals A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

2011 ◽  
Vol 48 (02) ◽  
pp. 420-434 ◽  
Author(s):  
Peter J. Thomas
Keyword(s):  
Lower Bound ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Condition ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Fire Model ◽  
Integrate And Fire ◽  
Ornstein Uhlenbeck Process ◽  
Time Density

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck processX(t) obeying dX= -βXdt+ σdWto reach a fixed threshold θ from a suprathreshold initial conditionx0&gt; θ &gt; 0 has a lower bound of the form ρ(t) &gt;kexp[-pe6βt] for positive constantskandpfor timestexceeding some positive valueu. We obtain explicit expressions fork,p, anduin terms of β, σ,x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

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 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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