A note on absorption probabilities in one-dimensional random walk via complex-valued martingales (Q2643381)

Let \(\{S_n: n\geq 0\}\) be a one-sided or two-sided random walk with a step taking values in a finite set of integers. For \(c>0\) and \(d<0\) define \(\tau:=\tau_{c,d}=\inf\{n\geq 0: S_n\leq d \;\;\text{or} \;\;S_n>c\}\). The authors characterize the set of all martingales of the form \(\{g(S_n):n\geq 0\}\), where \(g\) is a complex-valued function on integers. As an application of the result, an explicit form of probabilities \(P\{S_\tau=k\}\) is given which is computationally simpler than previously known formulae.