Continuity properties of optimal stopping value

Author:
John Elton

Journal:
Proc. Amer. Math. Soc. **105** (1989), 736-746

MSC:
Primary 60G40; Secondary 90C39

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).

**[AB]**Emil Artin and Hel Braun,*Introduction to algebraic topology*, Translated from the notes of Armin Thedy and Hel Braun by Erik Hemmingsen, Charles E. Merrill Publishing Co., Columbus, Ohio, 1969. MR**0247624****[B]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[CRS]**Y. S. Chow, Herbert Robbins, and David Siegmund,*Great expectations: the theory of optimal stopping*, Houghton Mifflin Co., Boston, Mass., 1971. MR**0331675****[CkD.]**David C. Cox and Robert P. Kertz,*Prophet regions and sharp inequalities for 𝑝th absolute moments of martingales*, J. Multivariate Anal.**18**(1986), no. 2, 242–273. MR**832998**, 10.1016/0047-259X(86)90072-2**[EK]**John Elton and Robert P. Kertz,*Comparison of stop rule and maximum expectations for finite sequences of exchangeable random variables*, Stochastic Anal. Appl.**9**(1991), no. 1, 1–23. MR**1096729**, 10.1080/07362999108809222**[HK]**T. P. Hill and Robert P. Kertz,*Additive comparisons of stop rule and supremum expectations of uniformly bounded independent random variables*, Proc. Amer. Math. Soc.**83**(1981), no. 3, 582–585. MR**627697**, 10.1090/S0002-9939-1981-0627697-3**[HK]**T. P. Hill and Robert P. Kertz,*Comparisons of stop rule and supremum expectations of i.i.d. random variables*, Ann. Probab.**10**(1982), no. 2, 336–345. MR**647508****[HK]**Theodore P. Hill and Robert P. Kertz,*Stop rule inequalities for uniformly bounded sequences of random variables*, Trans. Amer. Math. Soc.**278**(1983), no. 1, 197–207. MR**697070**, 10.1090/S0002-9947-1983-0697070-7**[K]**D. P. Kennedy,*Optimal stopping of independent random variables and maximizing prophets*, Ann. Probab.**13**(1985), no. 2, 566–571. MR**781423****[Ker]**Robert P. Kertz,*Stop rule and supremum expectations of i.i.d. random variables: a complete comparison by conjugate duality*, J. Multivariate Anal.**19**(1986), no. 1, 88–112. MR**847575**, 10.1016/0047-259X(86)90095-3**[KS]**Ulrich Krengel and Louis Sucheston,*Semiamarts and finite values*, Bull. Amer. Math. Soc.**83**(1977), no. 4, 745–747. MR**0436314**, 10.1090/S0002-9904-1977-14378-4**[KS]**Ulrich Krengel and Louis Sucheston,*On semiamarts, amarts, and processes with finite value*, Probability on Banach spaces, Adv. Probab. Related Topics, vol. 4, Dekker, New York, 1978, pp. 197–266. MR**515432****[P]**James Pickands III,*Extreme order statistics with cost of sampling*, Adv. in Appl. Probab.**15**(1983), no. 4, 783–797. MR**721706**, 10.2307/1427324

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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