Randomization in the first hitting time problem

In this paper, we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F(t)F(t), and a linear boundary, b(t)=μtb(t)=μt, find a distribution of the initial state such that the distribution of the first hitting time is F(t)F(t). This problem has important applications in credit risk modeling where the process represents the so-called distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.