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Continuity of trajectories of random walks with infinitesimal steps. (English) Zbl 0686.60069
Modeling and control of systems in engineering, quantum mechanics, economics, and biosciences, Proc. Bellman Continuum Workshop, Sophia Antipolis/Fr. 1988, Lect. Notes Control Inf. Sci. 121, 497-519 (1989).
[For the entire collection see Zbl 0668.00026.]
The authors attempt to explain to an audience not familiar with nonstandard analysis via Internal Set Theory (I.S.T) what Nelson means by continuous trajectories of a discrete diffusion process. This is done in somewhat generalized form. After a lengthy introduction to nonstandard analysis via I.S.T. they give a precise characterization for the sentence: “Almost every trajectory of such a discrete diffusion (or random walk) is continuous” in terms of S-continuity. However, it is not clear that these concepts are indeed accessible to an audience without the necessary intuitive background in nonstandard analysis.
Reviewer: R.A.Herrmann

MSC:
60G50 Sums of independent random variables; random walks
03H10 Other applications of nonstandard models (economics, physics, etc.)