An asymptotic expansion for the optimal stopping boundary in problems with non-linear costs of observation

The assumption of linear costs of observation usually leads to optimal stopping boundaries which are straight lines. For non-linear costs of observation, the question arises of how the shape of cost functions influences the shape of optimal stopping boundaries. In Irle (1987), (1988) it was shown that, under suitable assumptions on c, for the problem of optimal stopping (Wt + x)(+) - c(s + t), t is an element of [0, infinity), the optimal stopping boundary h(t) can be enscribed between k(1)/c'(t) and k(2)/c'(t) for some constants k(1), k(2). In this paper we find the exact asymptotic expansion h(t) = 1/(4c'(t))(1 + o(1)).