An Evaluation for the Probability Density of the First Hitting Time

Abstract

Let h(t) be a smooth function, B t a standard Brownian motion and t h =inf{ t ; B t = h ( t )} the first hitting time. In this paper, new formulations are derived to evaluate the probability density of the first hitting time. If u ( x , t ) denotes the density function of x= B t for t < t h , then u xx =2 u t and u ( h ( t ) ,t )=0 . Moreover, the hitting time density d h ( t ) is 1/2 u x ( h ( t ), t ) . Applying some partial differential equation techniques, we derive a simple integral equation for d h ( t ) . Two examples are demonstrated in this article.