A lower bound for the transition density function of a stochastic differential equation

1976 ◽  
Vol 4 (1) ◽  
pp. 143-149
Author(s):  
Abraham Boyarsky
Keyword(s):  
Differential Equation ◽  
Lower Bound ◽  
Stochastic Differential Equation ◽  
Density Function ◽  
Transition Density ◽  
Transition Density Function

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 Stochastic Volatility Effects on Correlated Log-Normal Random Variables

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Yong-Ki Ma
Keyword(s):  
Stochastic Process ◽  
Density Function ◽  
Process Model ◽  
Numerical Study ◽  
Correction Term ◽  
Order Term ◽  
Splitting Method ◽  
Transition Density ◽  
Analytic Approximation ◽  
Transition Density Function

The transition density function plays an important role in understanding and explaining the dynamics of the stochastic process. In this paper, we incorporate an ergodic process displaying fast moving fluctuation into constant volatility models to express volatility clustering over time. We obtain an analytic approximation of the transition density function under our stochastic process model. Using perturbation theory based on Lie–Trotter operator splitting method, we compute the leading-order term and the first-order correction term and then present the left and right skew scenarios through numerical study.

scholarly journals Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1363 ◽  
Author(s):  
Martynas Narmontas ◽  
Petras Rupšys ◽  
Edmundas Petrauskas
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Tree Height ◽  
Bivariate Distribution ◽  
General Purpose ◽  
Mixed Effect ◽  
Bivariate Probability

This study proposes a general bivariate stochastic differential equation model of population growth which includes random forces governing the dynamics of the bivariate distribution of size variables. The dynamics of the bivariate probability density function of the size variables in a population are described by the mixed-effect parameters Vasicek, Gompertz, Bertalanffy, and the gamma-type bivariate stochastic differential equations (SDEs). The newly derived bivariate probability density function and its marginal univariate, as well as the conditional univariate function, can be applied for the modeling of population attributes such as the mean value, quantiles, and much more. The models presented here are the basis for further developments toward the tree diameter–height and height–diameter relationships for general purpose in forest management. The present study experimentally confirms the effectiveness of using bivariate SDEs to reconstruct diameter–height and height–diameter relationships by using measurements obtained from mountain pine tree (Pinus mugo Turra) species dataset in Lithuania.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

1983 ◽  
Vol 20 (04) ◽  
pp. 754-765 ◽  
Author(s):  
Etsuo Isobe ◽  
Shunsuke Sato
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Series Expansion ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Hermite Expansion ◽  
Functional Series

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

1983 ◽  
Vol 20 (4) ◽  
pp. 754-765 ◽  
Author(s):  
Etsuo Isobe ◽  
Shunsuke Sato
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Series Expansion ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Hermite Expansion ◽  
Functional Series

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.

scholarly journals A Class of Solutions for the Hybrid Kinetic Model in the Tumor-Immune System Competition

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Carlo Cattani ◽  
Armando Ciancio
Keyword(s):  
Immune System ◽  
Kinetic Model ◽  
Density Function ◽  
Hybrid System ◽  
Kinetic Models ◽  
State Transition ◽  
Transition Density ◽  
Realistic Model ◽  
Transition Density Function ◽  
Competing Populations

In this paper, the hybrid kinetic models of tumor-immune system competition are studied under the assumption of pure competition. The solution of the coupled hybrid system depends on the symmetry of the state transition density which characterizes the probability of successful occurrences. Thus by defining a proper transition density function, the solutions of the hybrid system are explicitly computed and applied to a classical (realistic) model of competing populations.

scholarly journals A study on stochastic differential equation

2021 ◽  
Vol 5 (2) ◽  
pp. 68-75
Author(s):  
Govindaraju P ◽  
Senthil Kumar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Density Function ◽  
Discontinuous Drift ◽  
Point Of Departure

In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. In this paper we discussed The Euler-Maruyama method and this shows that a candidate density function based on the Euler-Maruyama method. The point of departure for this work is a particular SDE with discontinuous drift.

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