Israel Journal of Mathematics

, Volume 69, Issue 2, pp 173–192 | Cite as

Martingales with given maxima and terminal distributions

  • Robert P. Kertz
  • Uwe Rösler


Let μ be any probability measure onR with λ |x|dμ(x)<∞, and let μ* denote its associated Hardy and Littlewood maximal p.m. It is shown that for any p.m.v for which μ<ν<μ* in the usual stochastic order, there is a martingale (X t)0≦t≦1 for which sup0≦t≦1 X t andX 1 have respective p.m. 'sv and μ. The proof uses induction and weak convergence arguments; in special cases, explicit martingale constructions are given. These results provide a converse to results of Dubins and Gilat [6]; applications are made to give sharp martingale and ‘prophet’ inequalities.


Brownian Motion Probability Measure Induction Hypothesis Probability Space Weak Convergence 
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  1. 1.
    J. Azema and M. Yor,Une solution simple au probleme de Skorokhod, inSem Probab. XIII, Lecture Notes in Math. 721, Springer-Verlag, New York, 1979, pp. 90–115.CrossRefGoogle Scholar
  2. 2.
    J. Azema and M. Yor,Le probleme de Skorokhod: complements a l'expose precedent, inSem. Probab. XIII, Lecture Notes in Math. 721, Springer-Verlag, New York, 1979, pp. 625–633.Google Scholar
  3. 3.
    D. P. Bertsekas and S. E. Shreve,Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York, 1978.MATHGoogle Scholar
  4. 4.
    P. Billingsley,Convergence of Probability Measures, Wiley, New York, 1968.MATHGoogle Scholar
  5. 5.
    D. Blackwell and L. E. Dubins,A converse to the dominated convergence theorem, Illinois J. Math.7 (1963), 508–514.MATHMathSciNetGoogle Scholar
  6. 6.
    L. E. Dubins and D. Gilat,On the distribution of maxima of martingales, Proc. Am. Math. Soc.68 (1978), 337–338.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    L. E. Dubins and J. Pitman,A maximal inequality for skew fields, Z. Wahrscheinlichkeitstheor. Verw. Geb.52 (1980) 219–227.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. N. Ethier and T. G. Kurtz,Markov Processes Characterization and Convergence, Wiley, New York, 1986.MATHGoogle Scholar
  9. 9.
    D. Gilat,On the ratio of the expected maximum of a martingale and the L p-norm of its last term, Isr. J. Math.63 (1988), 270–280.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    G. H. Hardy and J. E. Littlewood,A maximal theorem with function theoretic applications, Acta. Math.54 (1930), 81–116.CrossRefMathSciNetGoogle Scholar
  11. 11.
    T. P. Hill and R. P. Kertz,Stop rule inequalities for uniformly bounded sequences of random variables, Trans. Am. Math. Soc.278 (1983), 197–207.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. D. Jacka,Doob's inequality revisited: a maximal H 1-embedding, Stoch. Processes Appl,29 (1988), 281–290.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    I. Karatzas and S. E. Shreve,Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988.MATHGoogle Scholar
  14. 14.
    U. Krengel and L. Sucheston,Semiamarts and finite values, Bull. Am. Math. Soc.83 (1977), 745–747.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    U. Krengel and L. Sucheston,On semiamarts, amarts, and processes with finite value, inAdvances in Probability, Vol. 4, Marcel Dekkar, New York, 1978, pp. 197–266.Google Scholar
  16. 16.
    P. A. Meyer,Probability and Potentials, Blaisdell Publ. Co., Waltham, Mass., 1966.MATHGoogle Scholar
  17. 17.
    E. Perkins,The Cereteli—Davis solution to the H 1-embedding problem and an optimal embedding in Brownian motion, inSeminar on Stochastic Processes, Birkhäuser, Boston, 1985.Google Scholar
  18. 18.
    V. Pestien,An extended Fatou equation and continuous-time gambling, Adv. Appl. Prob.14 (1982), 309–323.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. I. Resnick,Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987.MATHGoogle Scholar
  20. 20.
    G. R. Shorack and J. A. Wellner,Empirical Processes with Applications to Statistics, Wiley, New York, 1986.Google Scholar
  21. 21.
    D. Stoyan,Comparison Methods for Queues and Other Stochastic Models (D. J. Daley, ed.), Wiley, New York, 1983.Google Scholar
  22. 22.
    V. Strassen,The existence of probability measures with given marginals, Ann. Math. Statist.36 (1965), 423–439.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    D. P. van der Vecht,Inequalities for Stopped Brownian Motion, C. W. I. Tract 21, Mathematisch Centrum, Amsterdam, 1986.MATHGoogle Scholar

Copyright information

© Hebrew University 1990

Authors and Affiliations

  • Robert P. Kertz
    • 1
  • Uwe Rösler
    • 2
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Institut für Mathematische Statistik und WirtschaftsmathematikUniversität GöttingenGöttingenFRG

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