Henrik Bohnenkamp

 

Nonstandard probabilistic modeling

 

Abstract:

Non-standard mathematics is an alternative approach to deal with the
continuum. Most developed is non-standard analysis, which is not based
on limits, but on infinitely small numbers, the infinitesimals.  There
exists a well worked-out theory of non-standard stochastics. Rather
than using the real ``$x$-axis'' to model time, a so-called
hyper-finite set is used, which structurally is very similar to a
finite set of natural numbers (i.e. there is always a next bigger
number, up to the largest), but where the step-size is
infinitesimal. This approach has, for example, been used to describe a
random walk with infinitesimal small steps (``Brownian motion''),
which is a good example how to express a continuous model in a
quasi-discrete way.

In this presentation I will try to sketch the very basic notions of
nonstandard math, the possible applications to stochastic modeling as
we know it, and why a non-standard approach might provide a unifying
framework for many, if not all, stochastic and decision models known
so far.