Yes, the Martingale system is allowed in casinos. It is a betting strategy that involves doubling your bet after each loss in the hope of winning back your losses and then some.

The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice). It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.