Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics

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.

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

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.

A Stochastic Lomax Diffusion Process: Statistical Inference and Application

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.

Inference of Sojourn Time and Transition Density using the NLST X-ray Screening Data in Lung Cancer

Aims: The aim of this study is to provide statistical inference of the sojourn time and the transition probability from the disease free to the preclinical state of lung cancer for male and female smokers using lung cancer data from the National Lung Screening Trial (NLST). Materials and Methods: We applied a likelihood function to the lung cancer data, to obtain Bayesian inference of the transition probability and the sojourn time distribution. A log-normal distribution was used for the transition probability density function multiplied by 30%, and a Weibull distribution was used to model the sojourn time in the preclinical state. Results: The estimate of screening sensitivity is 0.61 for males and 0.62 for females. Early transition happened before age 50 and lasted until after age 90. The transition probability from the disease free to the preclinical state has a single maximum at around age 73 for males and 72 for females. For male, the Bayesian posterior mean, and median sojourn time are 1.33 and 1.27 years, respectively. For female, the corresponding posterior mean, and median sojourn time are 1.23 and 1.21 years, respectively. Conclusion: Our estimation showed that male smokers are more vulnerable to lung cancer, because they have a higher transition probability density than the same aged female smokers. The female smokers have a slightly shorter mean sojourn time than the male, meaning that they are quicker to develop clinical symptom of lung cancer.

Asymptotic expansion of the transition density of the semigroup associated to a SDE driven by Lévy noise

In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L ( t ) = L t , t > 0. The transition probability density p t , t > 0 of the semigroup associated to the solution u t , t ⩾ 0 of the SDE is given by a power series expansion. The series expansion of p t can be re-expressed in terms of Feynman graphs and rules. We will also prove that p t , t > 0 has an asymptotic expansion in power of a parameter β > 0, and it can be given by a convergent integral. A remark on some applications will be given in this work.

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

Over 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.

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

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.

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

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.

Неравновесная кинетика начальной стадии фазового перехода

Kinetic partial differential equations of Kolmogorov-Feller and Einstein-Smoluchowski equation with nonlinear coefficients are solved by a new, stable numerical methods. The theory of stochastic dynamic variables establishes the connection of the solution of Ito stochastic equations in the sense of Stratonovich for the trajectories of Wiener random processes with the transition probability density of these processes, or distribution functions of kinetic equations. The classical theory of nucleation (formation of nuclei of the first order phase transition) describes a non-equilibrium stage of the condensation process by a diffusion random process in the space of the size of the nuclei of the phase transition, when fluctuations affect the clustering of the nuclei. The model of formation of vacancy-gas defects (pores, blisters) in the crystal lattice, arising as a result of its irradiation by inert gas ions xenon, is supplemented by the consideration of Brownian motion of non-point lattice defects, occurring under the action of superposition of paired long-range potentials of indirect elastic interaction of pores between themselves and with the boundaries of the layers. Pores coordinates are changing at times of the order of 10 − 100 ms, sustainable algorithms for calculating which provide a self-consistent defnition spatial - temporal structures of porosity in the sample. According to calculations of 106 trajectories, non-equilibrium kinetic functions were found. Pores distribution in size and coordinates in the layers of the irradiated materials, they characterize the fluctuation instability the initial stage of the phase transition, they are estimated local stresses and porosity in the model volume.

On single-layer potentials for a class of pseudo-differential equations related to linear transformations of a symmetric $\alpha$-stable stochastic process

In this article an arbitrary invertible linear transformations of a symmetric $\alpha$-stable stochastic process in $d$-dimensional Euclidean space $\mathbb{R}^d$ are investigated. The result of such transformation is a Markov process in $\mathbb{R}^d$ whose generator is the pseudo-differential operator defined by its symbol $(-(Q\xi,\xi)^{\alpha/2})_{\xi\in\mathbb{R}^d}$ with some symmetric positive definite $d\times d$-matrix $Q$ and fixed exponent $\alpha\in(1,2)$. The transition probability density of this process is the fundamental solution of some parabolic pseudo-differential equation. The notion of a single-layer potential for that equation is introduced and its properties are investigated. In particular, an operator is constructed whose role in our consideration is analogous to that the gradient in the classical theory. An analogy to the classical theorem on the jump of the co-normal derivative of the single-layer potential is proved. This result can be applied for solving some boundary-value problems for the parabolic pseudo-differential equations under consideration. For $\alpha=2$, the process under consideration is a linear transformation of Brownian motion, and all the investigated properties of the single-layer potential are well known.