On the inverse first-passage-time problem for a Wiener process

The inverse first-passage problem for a Wiener process $(W_t{)}_{t\ge 0}$ seeks to determine a function $b:{\mathbb{R}}_{+}\to \mathbb{R}$ such that $$\tau =\inf \{t>0\mathrm{\mid }{W}_t\ge b(t)\}$$ has a given law. In this paper two methods for approximating the unknown function $b$ are presented. The errors of the two methods are studied. A set of examples illustrates the methods. Possible applications are enlighted.