Advertisement

Stochastic processes on the basis of new measure theory

  • Heinz KönigEmail author
Chapter
  • 1.1k Downloads

Abstract

The present article describes the reformation of certain basic structures, first in measure and integration as in the previous work of the author, and on this basis then in stochastic processes. Both times the aim is to overcome certain well-known substantial difficulties.

Keywords

Traditional and new stochastic processes their canonical and maximal measures their essential subsets the Wiener and Poisson processes inner premeasures and their maximal inner extensions projective limit theorems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Bauer, Wahrscheinlichkeitstheorie. 4th ed. de Gruyter 1991, English translation 1996.Google Scholar
  2. 2.
    N. Bourbaki, Intégration. Chap.1–4, 2ième ed. Hermann 1965, Chap.5, 2ème ed. Hermann 1967, Chap.IX, Hermann 1969, English translation Springer 2004.Google Scholar
  3. 3.
    C. Carathéodory, Über das lineare Mass von Punktmengen - eine Verallgemeinerung des Längenbegriffs. Nachr. K. Ges. Wiss. Göttingen, Math.-Nat. Kl. 1914, pp. 404–426. Reprinted in: Gesammelte Mathematische Schriften, Vol. IV, pp. 249–275. C.H. Beck 1956.Google Scholar
  4. 4.
    C. Dellacherie and P.A. Meyer, Probability and Potential. North-Holland 1978.Google Scholar
  5. 5.
    J. L. Doob, Probability in function spaces. Bull. Amer. Math. Soc. 53(1947), 15–30.MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. L. Doob, Stochastic Processes. Wiley 1953.Google Scholar
  7. 7.
    D.H. Fremlin, Measure Theory. Vol.1–4 Torres Fremlin 2000–2003 (in the references the first digit of an item indicates its volume). "http://www.essex.ac.uk/maths/staff/fremlin/mt.htm".
  8. 8.
    W. Hackenbroch and A. Thalmaier, Stochastische Analysis. Teubner 1994.Google Scholar
  9. 9.
    S. Kakutani, Notes on infinite product measure spaces II. Proc. Imp. Acad. Tokyo 19(1943), 184–188.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Kisyński, On the generation of tight measures. Studia Math. 30(1968), 141–151.MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Kolmogorov (= Kolmogoroff), Grundbegriffe derWahrscheinlichkeitsrechnung. Springer 1933, Reprint 1973.Google Scholar
  12. 12.
    H. König, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Springer 1997.Google Scholar
  13. 13.
    H. König, The product theory for inner premeasures. Note di Mat. 17(1997), 235–249.MathSciNetzbMATHGoogle Scholar
  14. 14.
    H. König, Measure and Integration: An attempt at unified systematization. Rend. Istit. Mat. Univ. Trieste 34(2002), 155–214. Preprint under "http://www.math.uni-sb.de/PREPRINTS/preprint42_OnlinePDF.pdf".
  15. 15.
    H. König, Projective limits via inner premeasures and the true Wiener measure. Mediterranean J. Math. 1(2004), 3–42. Preprint under "http://www.math.uni-sb.de/PREPRINTS/preprint83_OnlinePDF.pdf".
  16. 16.
    H. König, Stochastic processes in terms of inner premeasures. Note di Mat. Preprint under "http://www.math.uni-sb.de/PREPRINTS/preprint105_OnlinePDF.pdf".
  17. 17.
    H. König, The new maximal measures for stochastic processes. Z. Analysis Anwendungen. Preprint under "http://www.math.uni-sb.de/PREPRINTS/preprint117_OnlinePDF.pdf".
  18. 18.
    H. König, Essential sets and support sets for stochastic processes. Preprint under "http://www.math.uni-sb.de/PREPRINTS/preprint118_OnlinePDF.pdf".
  19. 19.
    E. Nelson, Regular probability measures on function spaces. Ann. Math. 69(1959), 630–643.MathSciNetCrossRefGoogle Scholar
  20. 20.
    K.R. Stromberg, Probability for Analysts. Chapman & Hall 1994.Google Scholar
  21. 21.
    T. Tjur, On the Mathematical Foundations of Probability. Lecture Notes 1, Inst. Math. Statist. Univ. Copenhagen 1972.Google Scholar
  22. 22.
    T. Tjur, Probability based on Radon Measures. Wiley 1980.Google Scholar
  23. 23.
    F. Topsøe, Topology and Measure. Lect. Notes Math. 133, Springer 1970.Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations