Markovian contact processes

1978 ◽  
Vol 10 (1) ◽  
pp. 85-108 ◽  
Author(s):  
Denis Mollison
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Birth Process ◽  
Linear Processes ◽  
Contact Processes ◽  
Spatial Spread ◽  
Non Linear ◽  
And Migration ◽  
Contact Distribution ◽  
Continuous Time Processes

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St(θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists.An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St/t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes.Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

Markovian contact processes

1978 ◽  
Vol 10 (01) ◽  
pp. 85-108 ◽  
Author(s):  
Denis Mollison
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Birth Process ◽  
Linear Processes ◽  
Contact Processes ◽  
Spatial Spread ◽  
Non Linear ◽  
And Migration ◽  
Contact Distribution ◽  
Continuous Time Processes

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St (θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists. An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St /t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes. Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

The Design of Optimum Controllers for Linear Processes With Energy Limitations

1963 ◽  
Vol 85 (2) ◽  
pp. 181-192 ◽  
Author(s):  
Bernard Friedland
Keyword(s):  
Energy Consumption ◽  
Discrete Time ◽  
Continuous Time ◽  
Analog Computer ◽  
Minimum Time ◽  
Linear Processes

A special-purpose analog computer which can be used as the optimum controller for linear processes with energy-limited control inputs is described. The computer may be used to force the process state to a desired state in minimum time; or to minimize the terminal error at a specified time; or to minimize the energy consumption, if the error can be reduced to zero in the specified time. Continuous-time and discrete-time processes are both considered.

scholarly journals Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Caishi Wang ◽  
Jinshu Chen
Keyword(s):  
Brownian Motion ◽  
Growth Condition ◽  
Discrete Time ◽  
Continuous Time ◽  
Orthonormal Basis ◽  
Continuous Time Processes ◽  
Characterization Theorems ◽  
Normal Martingale

We aim at characterizing generalized functionals of discrete-time normal martingales. LetM=(Mn)n∈Nbe a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals ofMwith an appropriate orthonormal basis forM’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.

scholarly journals A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

2004 ◽  
Vol 41 (03) ◽  
pp. 601-622 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Alexander Lindner ◽  
Ross Maller
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Levy Process ◽  
Volatility Models ◽  
Garch Processes ◽  
Continuous Time Processes ◽  
Necessary And Sufficient

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

NONPARAMETRIC NONSTATIONARITY TESTS

2013 ◽  
Vol 30 (1) ◽  
pp. 127-149 ◽  
Author(s):  
Federico M. Bandi ◽  
Valentina Corradi
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Nonlinear Process ◽  
Additive Functional ◽  
Finite Sample ◽  
Occupation Times ◽  
Discrete Time Case ◽  
Continuous Time Processes

We propose additive functional-based nonstationarity tests that exploit the different divergence rates of the occupation times of a (possibly nonlinear) process under the null of nonstationarity (stationarity) versus the alternative of stationarity (nonstationarity). We consider both discrete-time series and continuous-time processes. The discrete-time case covers Harris recurrent Markov chains and integrated processes. The continuous-time case focuses on Harris recurrent diffusion processes. Notwithstanding finite-sample adjustments discussed in the paper, the proposed tests are simple to implement and rely on tabulated critical values. Simulations show that their size and power properties are satisfactory. Our robustness to nonlinear dynamics provides a solution to the typical inconsistency problem between assumed linearity of a time series for the purpose of nonstationarity testing and subsequent nonlinear inference.

On Wald's equations in continuous time

1970 ◽  
Vol 7 (1) ◽  
pp. 59-68 ◽  
Author(s):  
W. J. Hall
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Stopping Time ◽  
Wiener Processes ◽  
Independent Increments ◽  
Randomly Stopped Sums ◽  
Continuous Time Processes ◽  
Wald's Identity ◽  
Stopped Sums ◽  
Inequality Theorem

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).

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