  In this paper we consider a connection between the famous Skorohod embedding
problem and the Shiryaev inverse problem for the first hitting time
distribution of a Brownian motion: given a probability distribution, $F$, find
a boundary such that the first hitting time distribution is $F$. By randomizing
the initial state of the process we show that the inverse problem becomes
analytically tractable. The randomization of the initial state allows us to
significantly extend the class of target distributions in the case of a linear
boundary and moreover allows us to establish connection with the Skorohod
embedding problem.
