Hitting Times with Respect to Time-Dependent Barriers

Abstract

In this short note, we prove that the distribution function of a hitting time of a Brownian Motion with respect to a differentiable, lower threshold solves a Volterra integral equation of second type. Moreover, we give an explicit expression for the kernel of the integral equation in terms of the boundary function and demonstrate how to find a solution of that equation (i.e. the distribution function) using Banach's Fixed-Point Theorem.As a second step, based on the first result, we find an integral representation of the terminal wealth distribution attached to the following, self-financing strategy: Let S_t be standard Geometric Brownian Motion stopped as soon as S_t hits a lower, time-dependent threshold. Then invest all of S_t in the money market account.