Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions

Abstract

We study the inverse boundary crossing problem for diffusions. Given a diffusion process X_t, and a survival distribution p on [0, ∞), we demonstrate that there exists a boundary b(t) such that p(t) = ℙ[τ > t], where τ is the first hitting time of X_t to the boundary b(t). The approach taken is analytic, based on solving a parabolic variational inequality to find b. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary b does indeed have p as its boundary crossing distribution. Since little is known regarding the regularity of b arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of X_t to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.