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

Integral Equations and the First Passage Time of Brownian Motions