Searching for a one-dimensional random walker

Let { x k } k ≧ − r be a simple Bernoulli random walk with x – r = 0. An integer valued threshold ϕ = {ϕ k } k ≧1 is called a search plan if |ϕ k +1 −ϕ k |≦1 for all k ≧ 1. If ϕ is a search plan let τ ϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ 1 = 0, the definition of ϕ does not depend on r , and the first crossing time τ ϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.