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Stopping times with given laws

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,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}).

This research was partially supported by the Danish Natural Science Council and by the U.S. National Science Foundation, Grant no. MCS76-07211.

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References

  • Halmos, P. (1950), Measure Theory (Princeton, Van Nostrand).

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© 1977 Springer-Verlag

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Dudley, R.M., Gutmann, S. (1977). Stopping times with given laws. In: Dellacherie, C., Meyer, P.A., Weil, M. (eds) Séminaire de Probabilités XI. Lecture Notes in Mathematics, vol 581. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087187

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  • DOI: https://doi.org/10.1007/BFb0087187

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08145-6

  • Online ISBN: 978-3-540-37383-4

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