scholarly journals The stationary probability density of a class of bounded Markov processes

2010 ◽  
Vol 42 (04) ◽  
pp. 986-993 ◽  
Author(s):  
Muhamad Azfar Ramli ◽  
Gerard Leng
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Markov Process ◽  
Probability Density ◽  
Markov Processes ◽  
Transition Probability ◽  
Stationary Probability ◽  
Probability Functions ◽  
Stationary Probability Density

In this paper we generalize a bounded Markov process, described by Stoyanov and Pacheco-González for a class of transition probability functions. A recursive integral equation for the probability density of these bounded Markov processes is derived and the stationary probability density is obtained by solving an equivalent differential equation. Examples of stationary densities for different transition probability functions are given and an application for designing a robotic coverage algorithm with specific emphasis on particular regions is discussed.

scholarly journals The stationary probability density of a class of bounded Markov processes

2010 ◽  
Vol 42 (4) ◽  
pp. 986-993 ◽  
Author(s):  
Muhamad Azfar Ramli ◽  
Gerard Leng
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Markov Process ◽  
Probability Density ◽  
Markov Processes ◽  
Transition Probability ◽  
Stationary Probability ◽  
Probability Functions ◽  
Stationary Probability Density

In this paper we generalize a bounded Markov process, described by Stoyanov and Pacheco-González for a class of transition probability functions. A recursive integral equation for the probability density of these bounded Markov processes is derived and the stationary probability density is obtained by solving an equivalent differential equation. Examples of stationary densities for different transition probability functions are given and an application for designing a robotic coverage algorithm with specific emphasis on particular regions is discussed.

Extension of Wald's first lemma to Markov processes

1999 ◽  
Vol 36 (01) ◽  
pp. 48-59 ◽  
Author(s):  
George V. Moustakides
Keyword(s):  
Integral Equation ◽  
Markov Process ◽  
Markov Processes ◽  
Correction Term ◽  
Stopping Time ◽  
Poisson Integral ◽  
Homogeneous Markov Process ◽  
Partial Sum ◽  
Poisson Integral Equation ◽  
Scalar Nonlinearity

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let S n denote the partial sum S n = θ(ξ1) + … + θ(ξ n ), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼S N = [lim n→∞𝔼θ(ξ n )]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξ N ). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξ N ) = o(𝔼N).

scholarly journals Parametrix construction for certain Lévy-type processes

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Victoria Knopova ◽  
Alexei Kulik
Keyword(s):  
Markov Process ◽  
Probability Density ◽  
Lower Bounds ◽  
Transition Probability ◽  
Local Operator ◽  
Upper And Lower Bounds ◽  
Transition Probability Density ◽  
Strong Markov Process ◽  
Non Local ◽  
Strong Markov

AbstractIn this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

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.

Order and Homogeneity of Family Specific Brand-switching Processes

1966 ◽  
Vol 3 (1) ◽  
pp. 48-54 ◽  
Author(s):  
William F. Massy
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Transition Probability ◽  
Empirical Work ◽  
Brand Choice ◽  
First Order ◽  
Brand Switching ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Crucial Assumption

Most empirical work on Markov processes for brand choice has been based on aggregative data. This article explores the validity of the crucial assumption that underlies such analyses, i.e., that all the families in the sample follow a Markov process with the same or similar transition probability matrices. The results show that there is a great deal of diversity among families’ switching processes, and that many of them are of zero rather than first order.

scholarly journals Dynamics of a Nonlinear Business Cycle Model Under Poisson White Noise Excitation

2015 ◽  
Vol 3 (2) ◽  
pp. 176-183 ◽  
Author(s):  
Jiaorui Li ◽  
Shuang Li
Keyword(s):  
Business Cycle ◽  
White Noise ◽  
Economic System ◽  
Probability Density ◽  
Stationary Probability ◽  
Cycle Model ◽  
Poisson White Noise ◽  
Stationary Probability Density ◽  
White Noise Excitation ◽  
Business Cycle Model

AbstractSeveral observations in real economic systems have shown the evidence of non-Gaussianity behavior, and one of mathematical models to describe these behaviors is Poisson noise. In this paper, stationary probability density of a nonlinear business cycle model under Poisson white noise excitation has been studied analytically. By using the stochastic averaged method, the approximate stationary probability density of the averaged generalized FPK equations are obtained analytically. The results show that the economic system occurs jump and bifurcation when there is a Poisson impulse existing in the periodic economic system. Furthermore, the numerical solutions are presented to show the effectiveness of the obtained analytical solutions.

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