A bound for the distribution of a stopping time for a stochastic system
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.
scientific article; zbMATH DE number 1023922