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Continuity properties of optimal stopping value


Author: John Elton
Journal: Proc. Amer. Math. Soc. 105 (1989), 736-746
MSC: Primary 60G40; Secondary 90C39
DOI: https://doi.org/10.1090/S0002-9939-1989-0949876-5
Erratum: Proc. Amer. Math. Soc. 107 (1989), 857.
MathSciNet review: 949876
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Abstract | References | Similar Articles | Additional Information

Abstract: The optimal stopping value of a sequence (finite or infinite) of integrable random variables is lower semicontinuous for the topology of convergence in distribution, when restricted to a collection with uniformly integrable negative parts. It is continuous for finite sequences which are adapted by a continuous invertible "triangular" function to independent sequences, such as partial averages; this is our main result. The proof depends on conditional weak convergence, uniform on compact sets, for such processes. A topological result on the inverses of triangular functions on iteratively connected domains may be of independent interest (§3).


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  • [AB] E. Artin and H. Braun, Introduction to algebraic topology, Charles Merrill, Columbus, Ohio, 1969. MR 0247624 (40:888)
  • [B] P. Billingsley, Convergence of probability measures, Wiley, New York, 1968. MR 0233396 (38:1718)
  • [CRS] Y. Chow, H. Robbins, and D. Siegmund, Great expectations: The theory of optimal stopping, Houghton Mifflin, Boston, 1971. MR 0331675 (48:10007)
  • [CkD.] Cox and R. Kertz, Prophet regions and sharp inequalities for $ p$-th absolute moments of martingales, J. Multivariate Anal. 18 (1986), 242-273. MR 832998 (87h:60044)
  • [EK] J. Elton and R. Kertz, Comparison of stop rule and maximum expectations for finite sequences of exchangeable random variables, (preprint). MR 1096729 (92b:60043)
  • [HK$ _{1}$] T. Hill and R. Kertz, Additive comparisons of stop rule and supremum expectations of uniformly bounded random variables, Proc. Amer. Math. Soc. 83 (1981), 582-585. MR 627697 (82j:60071)
  • [HK$ _{2}$] -, Comparisons of stop rule and supremum expectations of i.i.d. random variables, Ann. Prob. 10 (1982), 336-345. MR 647508 (83g:60053)
  • [HK$ _{3}$] -, Stop rule inequalities for uniformly bounded sequences of random variables, Trans. Amer. Math. Soc. 278 (1983), 197-207. MR 697070 (84i:60062)
  • [K] D. Kennedy, Optimal stopping of independent random variables and maximizing prophets, Ann. Probab. 13 (1985), 566-571. MR 781423 (86i:60128)
  • [Ker] R. Kertz, Stop rule and supremum expectations of i.i.d. random variables: a complete comparison by conjugate duality, J. Multivariate Anal. 19 (1986), 88-112. MR 847575 (87m:60102)
  • [KS$ _{1}$] U. Krengel and L. Sucheston, Semiamarts and finite values, Bull. Amer. Math. Soc. 83 (1977), 745-747. MR 0436314 (55:9261)
  • [KS$ _{2}$] -, On semiamarts, amarts, and processes with finite value, Adv. Prob. Related. Topics 4 (1978), 197-266. MR 515432 (80g:60053)
  • [P] J. Pickands, Extreme order statistics with cost of sampling, Adv. in Appl. Probab. 15 (1983), 783-797. MR 721706 (85c:60064)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0949876-5
Keywords: Optimal stopping value, weak convergence, conditional distribution, triangular function, adapted, iteratively connected
Article copyright: © Copyright 1989 American Mathematical Society

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