scholarly journals Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1879
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Transition Probability ◽  
Death Process ◽  
Continuous Approximation ◽  
Periodic Functions ◽  
Transition Probability Density ◽  
Asymptotic Behaviors ◽  
Conditional Moments ◽  
Intensity Functions ◽  
Random Fluctuations

The time-inhomogeneous Feller-type diffusion process, having infinitesimal drift α(t)x+β(t) and infinitesimal variance 2r(t)x, with a zero-flux condition in the zero-state, is considered. This process is obtained as a continuous approximation of a birth-death process with immigration. The transition probability density function and the related conditional moments, with their asymptotic behaviors, are determined. Special attention is paid to the cases in which the intensity functions α(t), β(t), r(t) exhibit some kind of periodicity due to seasonal immigration, regular environmental cycles or random fluctuations. Various numerical computations are performed to illustrate the role played by the periodic functions.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (2) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

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 Stochastic Lomax Diffusion Process: Statistical Inference and Application

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 100
Author(s):  
Ahmed Nafidi ◽  
Ilyasse Makroz ◽  
Ramón Gutiérrez Sánchez
Keyword(s):  
Diffusion Process ◽  
Density Function ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Trend Function ◽  
Non Linear Equation ◽  
Proposed Model ◽  
Adolescent Fertility ◽  
Trend Functions ◽  
Annealing Method

In this paper, we discuss a new stochastic diffusion process in which the trend function is proportional to the Lomax density function. This distribution arises naturally in the studies of the frequency of extremely rare events. We first consider the probabilistic characteristics of the proposed model, including its analytic expression as the unique solution to a stochastic differential equation, the transition probability density function together with the conditional and unconditional trend functions. Then, we present a method to address the problem of parameter estimation using maximum likelihood with discrete sampling. This estimation requires the solution of a non-linear equation, which is achieved via the simulated annealing method. Finally, we apply the proposed model to a real-world example concerning adolescent fertility rate in Morocco.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (02) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

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 Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 588 ◽  
Author(s):  
Eva María Ramos-Ábalos ◽  
Ramón Gutiérrez-Sánchez ◽  
Ahmed Nafidi
Keyword(s):  
Diffusion Process ◽  
Statistical Inference ◽  
Diffusion Processes ◽  
Transition Probability ◽  
Estimation Method ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Transition Probability Density ◽  
Ergodic Properties ◽  
New Family

In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data.

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