  The first passage time problem for Brownian motions hitting a barrier has
been extensively studied in the literature. In particular, many incarnations of
integral equations which link the density of the hitting time to the equation
for the barrier itself have appeared. Most interestingly, Peskir(2002b)
demonstrates that a master integral equation can be used to generate a
countable number of new equations via differentiation or integration by parts.
In this article, we generalize Peskir's results and provide a more powerful
unifying framework for generating integral equations through a new class of
martingales. We obtain a continuum of Volterra type integral equations of the
first kind and prove uniqueness for a subclass. Furthermore, through the
integral equations, we demonstrate how certain functional transforms of the
boundary affect the density function. Finally, we demonstrate a fundamental
connection between the Volterra integral equations and a class of Fredholm
integral equations.
