Gambler’s Ruin: The Duration of Play

We study the gambler’s ruin problem with a general distribution of the payoffs in each game. Assuming the expected value of the payoff distribution is negative, so that eventual ruin occurs with probability 1, we are interested in the distribution of the duration to ruin, also known as the first-passage time distribution. A generating function for this distribution is obtained. Exact expressions for the expected value and variance of this distribution, as well as asymptotic expressions for the case of large initial wealth, are derived.