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.

scholarly journals The Stochastic Modified Lundqvist-Korf Diffusion Process: Statistical and Computational Aspects and Application to Modeling of the CO2 Emission in Morocco

2021 ◽  
Author(s):  
Ahmed Nafidi ◽  
Abdenbi El azri ◽  
Ramón Gutiérrez Sanchez
Keyword(s):  
Diffusion Process ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Trend Function ◽  
Non Linear Equation ◽  
New Process ◽  
Computational Aspects ◽  
Exogenous Factors ◽  
Trend Functions

Abstract The main goal of this paper is to study the possibility of using a stochastic non-homogeneous (without exogenous factors) diffusion process to model the evolution of CO2 emissions in Morocco and concretely using a new process, in which the trend function is proportional to the modified Lundqvist-Korf growth curve. First, the main characteristics of the process are studied, then we establish a computational statistical methodology based on the maximum likelihood estimation method and the trend functions. When we are estimating the parameters of the process, a non-linear equation is obtained and the simulated annealing method is proposed to solve it after bounding the parametric space by a stagewise procedure. Also, to validate this methodology, we include the results obtained from several examples of simulation. Finally, the process and the methodology established are applied to real data corresponding to the evolution of CO2 emissions in Morocco.

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 Numerical simulation of the two-locus Wright-Fisher stochastic differential equation with application to approximating transition probability densities

2020 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Genetic Recombination ◽  
Transition Probability ◽  
Superior Performance ◽  
Transition Probability Density

AbstractOver the past decade there has been an increasing focus on the application of the Wright-Fisher diffusion to the inference of natural selection from genetic time series. A key ingredient for modelling the trajectory of gene frequencies through the Wright-Fisher diffusion is its transition probability density function. Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time, which presents opportunities for investigating natural selection while accounting for genetic recombination and local linkage. However, most existing methods for computing the transition probability density function of the Wright-Fisher diffusion are only applicable to one-locus problems. To address two-locus problems, in this work we propose a novel numerical scheme for the Wright-Fisher stochastic differential equation of population dynamics under natural selection at two linked loci. Our key innovation is that we reformulate the stochastic differential equation in a closed form that is amenable to simulation, which enables us to avoid boundary issues and reduce computational costs. We also propose an adaptive importance sampling approach based on the proposal introduced by Fearnhead (2008) for computing the transition probability density of the Wright-Fisher diffusion between any two observed states. We show through extensive simulation studies that our approach can achieve comparable performance to the method of Fearnhead (2008) but can avoid manually tuning the parameter ρ to deliver superior performance for different observed states.

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 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.

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