467 research outputs found
A Family of non-Gaussian Martingales with Gaussian Marginals
We construct a family of non-Gaussian martingales the marginals of which are
all Gaussian. We give the predictable quadratic variation of these processes
and show they do not have continuous paths. These processes are Markovian and
inhomogeneous in time, and we give their infinitesimal generators. Within this
family we find a class of piecewise deterministic pure jump processes and
describe the laws of jumps and times between the jumps.Comment: 16 pages, 2 figure
Branching Processes: Their Role in Epidemiology
Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large
Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities
Many populations, e.g. of cells, bacteria, viruses, or replicating DNA
molecules, start small, from a few individuals, and grow large into a
noticeable fraction of the environmental carrying capacity . Typically, the
elements of the initiating, sparse set will not be hampering each other and
their number will grow from in a branching process or Malthusian
like, roughly exponential fashion, , where is the size at
discrete time , is the offspring mean per individual (at the
low starting density of elements, and large ), and a sum of i.i.d.
random variables. It will, thus, become detectable (i.e. of the same order as
) only after around generations, when its density will
tend to be strictly positive. Typically, this entity will be random, even if
the very beginning was not at all stochastic, as indicated by lower case ,
due to variations during the early development. However, from that time
onwards, law of large numbers effects will render the process deterministic,
though initiated by the random density at time log , expressed through the
variable . Thus, acts both as a random veil concealing the start and a
stochastic initial value for later, deterministic population density
development. We make such arguments precise, studying general density and also
system-size dependent, processes, as . As an intrinsic size
parameter, may also be chosen to be the time unit. The fundamental ideas
are to couple the initial system to a branching process and to show that late
densities develop very much like iterates of a conditional expectation
operator.Comment: presented at IV Workshop on Branching Processes and their
Applications at Badajoz, Spain, 10-13 April, 201
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