On the first time that an Ito process hits a barrier

Abstract

This work deals with first hitting time densities of Ito processes whose local drift can be modeled in terms of a solution to Burgers equation. In particular, we derive the densities of the first time that these processes reach a moving boundary. We distinguish two cases: (a) the case in which the process has unbounded state space before absorption, and (b) the case in which the process has bounded state space before absorption. The reason as to why this distinction has to be made will be clarified. Next, we classify processes whose local drift can be expressed as a linear combination to solutions of Burgers equation. For example the local drift of a Bessel process of order 5 can be modeled as the sum of two solutions to Burgers equation and thus will be classified as of class $\mathcal{B}^2$. Alternatively, the Bessel process of order 3 has a local drift that can be modeled as a solution to Burgers equation and thus will be classified as of class $\mathcal{B}^1$. Examples of diffusions within class $\mathcal{B}^1$, and hence those to which the results described within apply, are: Brownian motion with linear drfit, the 3D Bessel process, the 3D Bessel bridge, and the Brownian bridge.