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A continuity question of Dubins and Savage

Part of: Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  22 June 2017

R. Laraki  and
W. Sudderth
R. Laraki*
Affiliation:
Université Paris-Dauphine
W. Sudderth*
Affiliation:
University of Minnesota
*
* Postal address: Director of Research at CNRS, Université Paris-Dauphine, PSL Research University, Lamsade, 75016 Paris, France.
** Postal address: School of Statistics, University of Minnesota, Minneapolis, MN 55455, USA. Email address: [email protected]
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Abstract

Lester Dubins and Leonard Savage posed the question as to what extent the optimal reward function U of a leavable gambling problem varies continuously in the gambling house Γ, which specifies the stochastic processes available to a player, and the utility function u, which determines the payoff for each process. Here a distance is defined for measurable houses with a Borel state space and a bounded Borel measurable utility. A trivial example shows that the mapping Γ ↦ U is not always continuous for fixed u. However, it is lower semicontinuous in the sense that, if Γn converges to Γ, then lim inf Un ≥ U. The mapping u ↦ U is continuous in the supnorm topology for fixed Γ, but is not always continuous in the topology of uniform convergence on compact sets. Dubins and Savage observed that a failure of continuity occurs when a sequence of superfair casinos converges to a fair casino, and queried whether this is the only source of discontinuity for the special gambling problems called casinos. For the distance used here, an example shows that there can be discontinuity even when all the casinos are subfair.

Keywords

Gambling theoryMarkov decision theory convergence of value functions

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 90C40: Markov and semi-Markov decision processes 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 462 - 473
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Dubins, L. and Meilijson, I. (1974). On stability for optimization problems. Ann. Prob. 2, 243–255. CrossRefGoogle Scholar
[2] Dubins, L. E. and Savage, L. J. (1965). How to Gamble If You Must. Inequalities for Stochastic Processes. McGraw-Hill, New York. Google Scholar
[3] Dubins, L., Maitra, A., Purves, R. and Sudderth, W. (1989). Measurable, nonleavable gambling problems. Israel J. Math. 67, 257–271. CrossRefGoogle Scholar
[4] Karatzas, I. and Sudderth, W. D. (1999). Control and stopping of a diffusion process on an interval. Ann. Appl. Prob. 9, 188–196. CrossRefGoogle Scholar
[5] Karatzas, I. and Wang, H. (2000). Utility maximization with discretionary stopping. SIAM J. Control Optimization 39, 306–329. CrossRefGoogle Scholar
[6] Karatzas, I. and Zamfirescu, I.-M. (2006). Martingale approach to stochastic control with discretionary stopping. Appl. Math. Optimization 53, 163–184. CrossRefGoogle Scholar
[7] Langen, H-J. (1981). Convergence of dynamic programming models. Math. Operat. Res. 6, 493–512. CrossRefGoogle Scholar
[8] Laraki, R. and Sudderth, W. D. (2004). The preservation of continuity and Lipschitz continuity by optimal reward operators. Math. Operat. Res. 29, 672–685. CrossRefGoogle Scholar
[9] Maitra, A. P. and Sudderth, W. D. (1996). Discrete Gambling and Stochastic Games. Springer, New York. CrossRefGoogle Scholar
[10] Pestien, V. C. and Sudderth, W. D. (1985). Continuous-time red and black: how to controla diffusion to a goal. Math. Operat. Res. 10, 599–611. CrossRefGoogle Scholar
[11] Pestien, V. C. and Sudderth, W. D. (1988). Continuous-time casino problems. Math. Operat. Res. 13, 364–376. CrossRefGoogle Scholar
[12] Strauch, R. E. (1967). Measurable gambling houses. Trans. Amer. Math. Soc. 126, 64–72. CrossRefGoogle Scholar
[13] Sudderth, W. D. (1969). On the existence of good stationary strategies. Trans. Amer. Math. Soc. 135, 399–414. CrossRefGoogle Scholar