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Stopping times with given laws

  • R. M. Dudley
  • Sam Gutmann
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 581)

Abstract

Given a stochastic process Xt, t ∈ T ⊂ R, and s ∈ R, then a) iff b): a) For every probability measure μ on [s, ∞], there is a stopping time τ for Xt with law L(τ)=μ; b) If At is the smallest σ-algebra for which Xu are mesurable for all u≤t, then P restricted to At is nonatomic for all t>s.

This note began with a question of G. Shiryaev, connected with the following example. Let Wt be a standard Wiener process, t ∈ T=[0, ∞]. Any exponential distribution on [0, ∞] will be shown to be the law of a stopping time. Using this, one can obtain a standard Poisson process Pt from Wt by a non-anticipating transformation, Pt=g({Xs: s≤t}).

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References

  1. Halmos, P. (1950), Measure Theory (Princeton, Van Nostrand).Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • R. M. Dudley
  • Sam Gutmann

There are no affiliations available

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