Diffusion approximations to linear stochastic difference equations with stationary coefficients

1977 ◽  
Vol 14 (1) ◽  
pp. 58-74 ◽  
Author(s):  
Harry A. Guess ◽  
John H. Gillespie
Keyword(s):  
Difference Equations ◽  
Time Scale ◽  
Population Biology ◽  
Correction Term ◽  
Random Environments ◽  
Diffusion Approximations ◽  
Uhlenbeck Process ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Approach Zero

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

Diffusion approximations to linear stochastic difference equations with stationary coefficients

1977 ◽  
Vol 14 (01) ◽  
pp. 58-74 ◽  
Author(s):  
Harry A. Guess ◽  
John H. Gillespie
Keyword(s):  
Difference Equations ◽  
Time Scale ◽  
Population Biology ◽  
Correction Term ◽  
Random Environments ◽  
Diffusion Approximations ◽  
Uhlenbeck Process ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Approach Zero

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

scholarly journals Examination of Particle Tails

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Tim Blackwell ◽  
Dan Bratton
Keyword(s):  
Power Law ◽  
Difference Equations ◽  
Particle Swarm ◽  
Second Order ◽  
Particle Swarm Optimisation ◽  
Position Distribution ◽  
Length Scales ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Second Order Equations

The tail of the particle swarm optimisation (PSO) position distribution at stagnation is shown to be describable by a power law. This tail fattening is attributed to particle bursting on all length scales. The origin of the power law is concluded to lie in multiplicative randomness, previously encountered in the study of first-order stochastic difference equations, and generalised here to second-order equations. It is argued that recombinant PSO, a competitive PSO variant without multiplicative randomness, does not experience tail fattening at stagnation.

scholarly journals On a first-order semipositone boundary value problem on a time scale

2014 ◽  
Vol 8 (2) ◽  
pp. 269-287
Author(s):  
Christopher Goodrich
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Boundary Value ◽  
First Order ◽  
General Time

We consider the existence of a positive solution to the first-order dynamic equation y?(t)+p(t)y?(t) = ?f (t, y?(t)), t?(a, b)T, subject to the boundary condition y(a) = y(b) + ?T1,T2 F(s, y(s)) ?s for ?1,?2 ? [a,b]T. In this setting, we allow f to take negative values for some (t; y). Our results generalize some recent results for this class of problems, and because we treat the problem on a general time scale T we provide new results for this problem in the case of differential, difference, and q-difference equations. We also provide some discussion of the applicability of our results.

Linear Stochastic Difference Equations

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Time Series ◽  
Difference Equation ◽  
Difference Equations ◽  
Martingale Difference ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
Order Linear ◽  
Economic Agents ◽  
First Order ◽  
Competitive Equilibria

This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted the adoption of economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a “martingale difference sequence” are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series.

A limit theorem for discrete-parameter random evolutions

1984 ◽  
Vol 16 (02) ◽  
pp. 281-292
Author(s):  
Gilles A. Blum
Keyword(s):  
Limit Theorem ◽  
Difference Equations ◽  
Diffusion Approximations ◽  
Stochastic Difference Equations ◽  
Markovian Environment ◽  
Discrete Parameter ◽  
Fisher Model ◽  
Random Evolutions

In this paper we obtain a limit theorem for discrete-parameter random evolutions. This theorem is then used to obtain diffusion approximations to the Wright-Fisher model in a Markovian environment and to sequences of stochastic difference equations.

A limit theorem for discrete-parameter random evolutions

1984 ◽  
Vol 16 (2) ◽  
pp. 281-292
Author(s):  
Gilles A. Blum
Keyword(s):  
Limit Theorem ◽  
Difference Equations ◽  
Diffusion Approximations ◽  
Stochastic Difference Equations ◽  
Markovian Environment ◽  
Discrete Parameter ◽  
Fisher Model ◽  
Random Evolutions

In this paper we obtain a limit theorem for discrete-parameter random evolutions. This theorem is then used to obtain diffusion approximations to the Wright-Fisher model in a Markovian environment and to sequences of stochastic difference equations.

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