scholarly journals Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 555 ◽  
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Time Dependent ◽  
Transition Probability Density ◽  
Special Boundaries ◽  
Special Cases ◽  
Regulation Functions

We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.

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.

scholarly journals On the Simulation of a Special Class of Time-Inhomogeneous Diffusion Processes

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 818
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Probability Density ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Probability Density Functions ◽  
Periodic Functions ◽  
Initial State ◽  
Transition Probability Density ◽  
Inhomogeneous Diffusion

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed.

scholarly journals On first-passage-time and transition densities for strongly symmetric diffusion processes

1997 ◽  
Vol 145 ◽  
pp. 143-161 ◽  
Author(s):  
A. Di Crescenzo ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Process ◽  
Closed Form ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
Closed Form Expression ◽  
First Passage ◽  
One Dimensional ◽  
Transition Functions

One dimensional diffusion processes have been increasingly invoked to model a variety of biological, physical and engineering systems subject to random fluctuations (cf., for instance, Blake, I. F. and Lindsey, W. C. [2], Abrahams, J. [1], Giorno, V. et al [10] and references therein). However, usually the knowledge of the ‘free’ transition probability density function (pdf) is not sufficient; one is thus led to the more complicated task of determining transition functions in the presence of preassigned absorbing boundaries, or first-passage-time densities for time-dependent boundaries (see, for instance, Daniels, H. E. [6], [7], Giorno, V. et al. [10]). Such densities are known analytically only in some special instances so that numerical methods have to be implemented in general (cf., for instance, Buono-core, A. et al [3], [4], Giorno, V. et al [11]). The analytical approach becomes particularly effective when the diffusion process exhibits some special features, such as the symmetry of its transition pdf. For instance, in [10] special symmetry conditions on the transition pdf of one-dimensional time-homogeneous diffusion process with natural boundaries are investigated to derive closed form results concerning the transition pdf’s and the first-passage-time pdf for particular time-dependent boundaries. On the other hand, by using the method of images, in [6] Daniels has obtained a closed form expression for the transition pdf of the standard Wiener process in the presence of a particular time-dependent absorbing boundary. It is interesting to remark that such density cannot be obtained via the methods described in [10], even though the considered process exhibits the kind of symmetry discussed therein.

scholarly journals First Passage Time Moments of Jump-Diffusions with Markovian Switching

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Jun Peng ◽  
Zaiming Liu
Keyword(s):  
Integral Equation ◽  
Recurrence Relations ◽  
Financial Models ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Markovian Switching ◽  
First Passage ◽  
Transition Probability Density ◽  
Jump Diffusions

Using an integral equation associated with generalized backward Kolmogorov's equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of jump-diffusions with Markovian switching. The results are used to find the expectation of first passage time of some financial models.

Mean first-passage times of Brownian motion and related problems

1952 ◽  
Vol 211 (1106) ◽  
pp. 431-443 ◽  
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Analytical Relationship ◽  
First Passage ◽  
Special Boundaries ◽  
Mean First Passage Times ◽  
Special Cases ◽  
Passage Times

The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.

On the evaluation of first-passage-time probability densities via non-singular integral equations

1989 ◽  
Vol 21 (1) ◽  
pp. 20-36 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Singular Integral Equations ◽  
Passage Time ◽  
Time Dependent ◽  
Computational Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

A new integral equation for the evaluation of first-passage-time probability densities

1987 ◽  
Vol 19 (04) ◽  
pp. 784-800 ◽  
Author(s):  
A. Buonocore ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Continuous Functions ◽  
Time Dependent ◽  
First Passage ◽  
One Dimensional ◽  
Probability Densities

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

A new integral equation for the evaluation of first-passage-time probability densities

1987 ◽  
Vol 19 (4) ◽  
pp. 784-800 ◽  
Author(s):  
A. Buonocore ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Continuous Functions ◽  
Time Dependent ◽  
First Passage ◽  
One Dimensional ◽  
Probability Densities

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

On the evaluation of first-passage-time probability densities via non-singular integral equations

1989 ◽  
Vol 21 (01) ◽  
pp. 20-36 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Singular Integral Equations ◽  
Passage Time ◽  
Time Dependent ◽  
Computational Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

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