jouer zebet

Let X_t+1 = X_t b_t where +b_t and -b_t occur with equal probability b_t is measurable !1_t, and the outcome b_t is measurable !1_t+1 (in other words, my "bet" b_t can only depend on what has happened so far and not on the future, but my knowledge !1_t includes the outcome of all past bets). Then {X_t|!1_t} is a martingale.

The Martingale System is commonly compared to betting in a casino with the hopes of breaking even. When a gambler who uses this method experiences a loss, they immediately double the size of the next bet. By repeatedly doubling the bet when they lose, the gambler, in theory, will eventually even out with a win.