Compound kernel estimates for the transition probability density of a Lévy process in $\mathbb R^n$

INTRINSIC COMPOUND KERNEL ESTIMATES FOR THE TRANSITION PROBABILITY DENSITY OF LÉVY-TYPE PROCESSES AND THEIR APPLICATIONSStarting with an integro-differential operator L, C2∞ ℜn, we prove that its C∞ℜn-closure is the generator of a Feller process X, which admits a transition probability density. To construct this transition probability density, we develop a version of the parametrix method and a verification procedure, which proves that the constructed object is the claimed one. As a part of the construction, we prove the intrinsic upper and lower estimates on the density. As an application of the constructed estimates we state the necessary and separately sufficient conditions under which a given Borel measure belongs to the Kato and Dynkin classes with respect to the constructed transition probability density.

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Parametrix construction for certain Lévy-type processes

Random Operators and Stochastic Equations ◽

10.1515/rose-2014-0032 ◽

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.

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

Journal of Applied Probability ◽

10.2307/3214570 ◽

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.

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

10.1101/2020.07.21.213769 ◽

2020 ◽

Cited By ~ 2

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.

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