On the Computation of the Survival Probability of Brownian Motion with Drift in a Closed Time Interval When the Absorbing Boundary Is a Step Function

Abstract

This paper provides explicit formulae for the probability that an arithmetic or a geometric Brownian motion will not cross an absorbing boundary defined as a step function during a finite time interval. Various combinations of downward and upward steps are handled. Numerical computation of the survival probability is done quasi-instantaneously and with utmost precision. The sensitivity of the survival probability to the number and the ordering of the steps in the boundary is analyzed.