On distibutions of first passage times of martingales arising in some gambling problems

Using a martingale technique we derive bounds and asymptotics for tail distributions of first passage times $$\tau _{b}$$ associated with crossing a level b for some gambling strategies. In particilar, for the case of martingale games with so-called “Oscar strategy” we show that $$P\{\tau _{b}>n\}\le C n^{-3/2}\,$$ for any level $$b>0$$ .