First passage time densities for random walk spans

A general expression is derived for the Laplace transform of the probability density of the first passage time for the span of a symmetric continuous-time random walk to reach level S. The authors show that when the mean time between steps is finite, the mean first passage time to S is proportional to S2. When the pausing time density is asymptotic to a stable density the authors show that the first passage density is also asymptotically stable. Finally, when the jump distribution of the random walk has an asymptotic form raised to the negative power alpha plus 1 with alpha restricted between the values of 0 and 2, it is shown that the mean first passage time to S goes like S raised to the corresponding power of alpha