Continuity properties of optimal stopping value

Elton, John

doi:10.1090/S0002-9939-1989-0949876-5

Continuity properties of optimal stopping value
HTML articles powered by AMS MathViewer

by John Elton PDF

Proc. Amer. Math. Soc. 105 (1989), 736-746 Request permission

Erratum: Proc. Amer. Math. Soc. 107 (1989), 857.

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).

References

Emil Artin and Hel Braun, Introduction to algebraic topology, Charles E. Merrill Publishing Co., Columbus, Ohio, 1969. Translated from the notes of Armin Thedy and Hel Braun by Erik Hemmingsen. MR 0247624
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Y. S. Chow, Herbert Robbins, and David Siegmund, Great expectations: the theory of optimal stopping, Houghton Mifflin Co., Boston, Mass., 1971. MR 0331675
David C. Cox and Robert P. Kertz, Prophet regions and sharp inequalities for $p$th absolute moments of martingales, J. Multivariate Anal. 18 (1986), no. 2, 242–273. MR 832998, DOI 10.1016/0047-259X(86)90072-2
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, DOI 10.1080/07362999108809222
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, DOI 10.1090/S0002-9939-1981-0627697-3
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
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, DOI 10.1090/S0002-9947-1983-0697070-7
D. P. Kennedy, Optimal stopping of independent random variables and maximizing prophets, Ann. Probab. 13 (1985), no. 2, 566–571. MR 781423
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, DOI 10.1016/0047-259X(86)90095-3
Ulrich Krengel and Louis Sucheston, Semiamarts and finite values, Bull. Amer. Math. Soc. 83 (1977), no. 4, 745–747. MR 436314, DOI 10.1090/S0002-9904-1977-14378-4
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
James Pickands III, Extreme order statistics with cost of sampling, Adv. in Appl. Probab. 15 (1983), no. 4, 783–797. MR 721706, DOI 10.2307/1427324