On the first passage time density of a continuous Martingale over a moving boundary

Abstract

In this paper we derive the density $\phi$ of the first time $T$ that a continuous martingale $M$ with non-random quadratic variation $<M{>}_{\cdot }:={\int }_0^{\cdot }{h}^2(u)du$ hits a moving boundary $f$ which is twice continuously differentiable, and $f^{\prime }/h\in {\mathbb{C}}^2[0,\infty )$. Thus, this work is an extension to case in which $M$ is in fact a one-dimensional standard Brownian motion $B$, as studied in Hernandez-del-Valle (2007).