A Multilevel birth-death particle system and its continuous diffusion

1993 ◽  
Vol 25 (03) ◽  
pp. 549-569 ◽  
Author(s):  
Yadong Wu
Keyword(s):  
Diffusion Process ◽  
Unique Solution ◽  
Diffusion Approximation ◽  
Particle System ◽  
Martingale Problem ◽  
The Moment ◽  
Birth Death

In this paper we introduce a multilevel birth-death particle system and consider its diffusion approximation which can be characterized as aM([R+)-valued process. The tightness of rescaled processes is proved and we show that the limitingM(R+)-valued process is the unique solution of theM([R+)-valued martingale problem for the limiting generator. We also study the moment structures of the limiting diffusion process.

A Multilevel birth-death particle system and its continuous diffusion

1993 ◽  
Vol 25 (3) ◽  
pp. 549-569 ◽  
Author(s):  
Yadong Wu
Keyword(s):  
Diffusion Process ◽  
Unique Solution ◽  
Diffusion Approximation ◽  
Particle System ◽  
Martingale Problem ◽  
The Moment ◽  
Birth Death

In this paper we introduce a multilevel birth-death particle system and consider its diffusion approximation which can be characterized as aM([R+)-valued process. The tightness of rescaled processes is proved and we show that the limitingM(R+)-valued process is the unique solution of theM([R+)-valued martingale problem for the limiting generator. We also study the moment structures of the limiting diffusion process.

scholarly journals The SEIS model, or, the contact process with a latent stage

2016 ◽  
Vol 53 (3) ◽  
pp. 783-801 ◽  
Author(s):  
Eric Foxall
Keyword(s):  
Particle System ◽  
Contact Process ◽  
Compartment Model ◽  
Spatial Dimension ◽  
Critical Values ◽  
System Model ◽  
Upper And Lower Bounds ◽  
Latent Stage ◽  
Latent Time ◽  
The Moment

AbstractThe susceptible→exposed→infectious→susceptible (SEIS) model is well known in mathematical epidemiology as a model of infection in which there is a latent period between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received no attention. For the particle system model in one spatial dimension, we give upper and lower bounds on the critical values, prove convergence of critical values in the limit of small and large latent time, and identify a limiting process to which the SEIS model converges in the limit of large latent time.

Plastic-Wave Propagation Effects in Transverse Impact of Membranes

1960 ◽  
Vol 27 (1) ◽  
pp. 172-176 ◽  
Author(s):  
B. Karunes ◽  
E. T. Onat
Keyword(s):  
Unique Solution ◽  
Average Rate ◽  
Initial Energy ◽  
Plane Motion ◽  
Middle Region ◽  
Bending Rigidity ◽  
Transverse Impact ◽  
Significant Information ◽  
Subsequent Motion ◽  
The Moment

The paper is concerned with the plane motion of a rigid-strain-hardening membrane attached to two parallel fixed supports. The membrane is subjected to a uniformly distributed transverse impulse and the subsequent motion of the membrane is to be determined with the particular emphasis on the variation of thickness in the final deflected shape. It is first shown that two essentially different initial modes of deformation exist depending on the average rate of hardening. For both modes, the analysis can be based on two types of waves of discontinuity until the moment when the compressive membrane forces occur in the middle region of the membrane. The presence of compressive forces will generally preclude the existence of a unique solution for further motion. The bending rigidity will probably have to be included into the analysis in order to obtain a unique solution. However, for the technically important rates of hardening and velocities, the kinetic energy of the membrane at the moment of occurrence of compressive forces is small compared with the initial energy, so that significant information could be obtained from the present analysis about the variation of thickness and hardening throughout the membrane.

scholarly journals The ruin problem for a Wiener process with state-dependent jumps

2020 ◽  
Vol 16 (1) ◽  
pp. 13-23
Author(s):  
M. Lefebvre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Jump Diffusion ◽  
Moment Generating Function ◽  
Continuous Part ◽  
State Dependent ◽  
The Moment ◽  
First Time

AbstractLet X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

Uniqueness for a Competing Species Model

1999 ◽  
Vol 51 (2) ◽  
pp. 372-448 ◽  
Author(s):  
Leonid Mytnik
Keyword(s):  
Unique Solution ◽  
Martingale Problem ◽  
Probability Measures ◽  
Competing Species ◽  
Uniqueness Of The Solution ◽  
Duality Technique

AbstractWe show that a martingale problem associated with a competing species model has a unique solution. The proof of uniqueness of the solution for the martingale problem is based on duality technique. It requires the construction of dual probability measures.

Analysis of transient behaviour of certain processes with return to a central state

1983 ◽  
Vol 20 (01) ◽  
pp. 61-70
Author(s):  
Peter G. Buckholtz ◽  
L. Lorne Campbell ◽  
Ross D. Milbourne ◽  
M. T. Wasan
Keyword(s):  
Probability Distribution ◽  
Diffusion Equation ◽  
Diffusion Approximation ◽  
Cash Management ◽  
Central State ◽  
Transient Behaviour ◽  
Management Problems ◽  
Transient Probability ◽  
Birth Death

In economics, cash management problems may be modelled by birth-death processes which reset to central states when a boundary is reached. The nature of the transient behaviour of the probability distribution of such processes symmetric about a central state is investigated. A diffusion approximation of such processes is given and the transient probability behaviour derived from the diffusion equation.

scholarly journals Filtering with a limiter

1998 ◽  
Vol 11 (3) ◽  
pp. 289-300 ◽  
Author(s):  
R. Liptser ◽  
P. Muzhikanov
Keyword(s):  
Weak Convergence ◽  
Diffusion Process ◽  
White Noise ◽  
Discrete Time ◽  
Diffusion Approximation ◽  
Conditional Expectation ◽  
Gaussian White Noise ◽  
Sampling Step ◽  
Non Gaussian ◽  
Filtering Problem

We consider a filtering problem for a Gaussian diffusion process observed via discrete-time samples corrupted by a non-Gaussian white noise. Combining the Goggin's result [2] on weak convergence for conditional expectation with diffusion approximation when a sampling step goes to zero we construct an asymptotic optimal filter. Our filter uses centered observations passed through a limiter. Being asymptotically equivalent to a similar filter without centering, it yields a better filtering accuracy in a prelimit case.

The spatial distribution of Tribolium confusum

1980 ◽  
Vol 17 (4) ◽  
pp. 895-911 ◽  
Author(s):  
Eric Renshaw
Keyword(s):  
Spatial Distribution ◽  
Diffusion Process ◽  
Tribolium Confusum ◽  
Lattice Points ◽  
Closed Container ◽  
Migration Model ◽  
Birth Death

Neyman, Park and Scott (1956) describe an experiment which they performed to determine the spatial distribution of Tribolium confusum developing within a closed container. To explain the concentration of beetles at the boundary a birth–death–migration model is developed in which the beetles may migrate over a set of lattice points, and this is shown to produce a distribution of the required shape. Not only is this distribution independent of the number of lattice points, but it is also indistinguishable from the associated diffusion process.

scholarly journals Notes on the birth–death prior with fossil calibrations for Bayesian estimation of species divergence times

2016 ◽  
Vol 371 (1699) ◽  
pp. 20150128 ◽  
Author(s):  
Mario dos Reis
Keyword(s):  
Conditional Density ◽  
Death Process ◽  
Divergence Times ◽  
Large Set ◽  
Sources Of Information ◽  
Fossil Calibration ◽  
The Times ◽  
The Moment ◽  
Different Sources ◽  
Birth Death

Constructing a multi-dimensional prior on the times of divergence (the node ages) of species in a phylogeny is not a trivial task, in particular, if the prior density is the result of combining different sources of information such as a speciation process with fossil calibration densities. Yang & Rannala (2006 Mol. Biol. Evol . 23, 212–226. ( doi:10.1093/molbev/msj024 )) laid out the general approach to combine the birth–death process with arbitrary fossil-based densities to construct a prior on divergence times. They achieved this by calculating the density of node ages without calibrations conditioned on the ages of the calibrated nodes. Here, I show that the conditional density obtained by Yang & Rannala is misspecified. The misspecified density can sometimes be quite strange-looking and can lead to unintentionally informative priors on node ages without fossil calibrations. I derive the correct density and provide a few illustrative examples. Calculation of the density involves a sum over a large set of labelled histories, and so obtaining the density in a computer program seems hard at the moment. A general algorithm that may provide a way forward is given. This article is part of the themed issue ‘Dating species divergences using rocks and clocks’.

Point-based polygonal models for random graphs

1993 ◽  
Vol 25 (2) ◽  
pp. 348-372 ◽  
Author(s):  
T. Arak ◽  
P. Clifford ◽  
D. Surgailis
Keyword(s):  
Computer Simulations ◽  
Random Graphs ◽  
Particle System ◽  
Two Dimensional ◽  
One Dimensional ◽  
Space And Time ◽  
Markov Random ◽  
Alternative Description ◽  
Polygonal Models ◽  
Birth Death

We define a class of two-dimensional Markov random graphs with I, V, T and Y-shaped nodes (vertices). These are termed polygonal models. The construction extends our earlier work [1]– [5]. Most of the paper is concerned with consistent polygonal models which are both stationary and isotropic and which admit an alternative description in terms of the trajectories in space and time of a one-dimensional particle system with motion, birth, death and branching. Examples of computer simulations based on this description are given.

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