2011 | OriginalPaper | Chapter
Epilogue: Genericity of Hyperfinite Loeb Path Spaces
Authors : Sergio Albeverio, Ruzong Fan, Frederik Herzberg
Published in: Hyperfinite Dirichlet Forms and Stochastic Processes
Publisher: Springer Berlin Heidelberg
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An important part of this book has been concerned with the investigation of hyperfinite Markov chains
$$ {X\,=\,(X_t)_{t\in T}}$$
on an internal probability space
Ω
. If we confine ourselves to a finite time horizon – say, 1 – and if we assume
$${\Delta {t}}\, $$
to be a reciprocal hypernatural number, then
$$ {H} \,=\,{1/{\Delta{t}}}\in\,{^*}\mathbb{N}\,\backslash\mathbb{N}\, $$
will be the number of possible transitions occurring between
$$ {0\,\in\,T}\,$$
and
$$ {1\,\in\,T\,.}\,$$
Because
X
was assumed to be a hyperfinite Markov chain starting at a given point x
0
in the state space, the number of possible paths between 0 and 1 will be hyperfinite. Thus, provided we are only interested in studying events that lie in the filtration generated by
X
, we may assume Ω to be hyperfinite.