Simulation of stopped diffusions

In this work, we study standard Euler updates for simulating stopped diffusions. As an immediate application, we discuss the computation of first exit times of diffusions from a domain. We focus on one-dimensional situations and show how the ideas for the simulation of killed diffusions can be adapted to this problem. In particular, we give a fully implementable algorithm to compute the first exit time from an interval numerically. The Brownian motion case is treated in detail and extensions to general diffusions are given. Special emphasis is given to numerical experiments: For every ansatz, we include numerical experiments confirming the conjectured accuracy of our methods. Our algorithm is of order one in a weak sense. Comparisons with other algorithms are shown. Results that are superior to those obtained with other methods are presented. When approximating a first hitting time distribution, the results obtained with our algorithm are much better than those achieved with other methods.