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

TL;DR

This paper derives the probability density function for the first hitting time of a continuous martingale with a specified quadratic variation to a moving boundary, extending previous Brownian motion results to more general martingales.

Contribution

It provides a new explicit formula for the first passage time density of a continuous martingale over a moving boundary, generalizing prior Brownian motion findings.

Findings

Derived the density function for first passage times of martingales.

Extended Brownian motion results to general continuous martingales.

Provided conditions on the boundary function for the density to be valid.

Abstract

In this paper we derive the density $\varphi$ 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 $f'/h\in\mathbb{C}^2[0,\infty)$. 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).