Simulation of Brownian motion at first-passage times

We show how to simulate Brownian motion not on a regular time grid, but on a regular spatial grid. That is, when it first hits points in δZ for some δ>0δ>0. Central to our method is an algorithm for the exact simulation of ττ, the first time Brownian motion hits ±1±1. This work is motivated by boundary hitting problems for time-changed Brownian motion, such as appear in mathematical finance when pricing barrier-options.