A bound for the distribution of a stopping time for a stochastic system
type: Item
instance of: scholarly article
title: A bound for the distribution of a stopping time for a stochastic system
zbMATH Open document ID: 0958.60043
DOI: 10.1007/BF02104661
published in: Siberian Mathematical Journal
publication date: 2001-04-09T00:00:00Z
review text: The so-called ``Oscar's system'' in gambling is a system aimed to win one betting unit with a high probability in a reasonably short time. This is achieved by increasing the bet size by one unit after each win. The system can be modeled by an irreducible Markov chain on the two-dimensional integer lattice, and was recently extensively studied by \textit{S. N. Ethier} [``Finding the edge: Mathematical and quantitative aspects of gambling'', in: Proc. 9th Internat. Conf. Gambling and Risk Taking. Vol. 4 (W. R. Eadington, and J. A. Cornelius, eds.) and Ann. Appl. Probab. 6, No. 4, 1248-1259 (1996; Zbl 0876.60051)], the most difficult and interesting case being the ``critical'' one when success probability \(p=1/2\). We suggest a new approach to analyse such systems which combines embedding techiques with boundary problems for the Wiener process. This approach allows one to get a rather tight bound for the distribution tail of the duration of the game, and can be extended to more general models of random walks with varying jump sizes.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 91A60
zbMATH DE Number: 1023922
zbMATH Keywords: gambling system
zbMATH Keywords: embedding
zbMATH Keywords: boundary problems for Wiener process
author: K. Borovkov
MaRDI profile type: Publication
cites work: A gambling system and a Markov chain
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description: scientific article; zbMATH DE number 1023922