Randomization in the Default Boundary Problem
by Ken Jackson of the University of Toronto,
February 12, 2008
Abstract: 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), and a linear boundary, b(t)= μt, find a distribution of the initial state such that the distribution of the first hitting time is F(t). This problem has important applications in credit risk modeling where the process represents, 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.
Keywords: First hitting time, default boundary problem, credit risk.