Computing exit times with the Euler scheme

In this note we study standard Euler updates for computing first exit times of general diffusions from a domain. We focus on one dimensional situations and show how the ideas of Mannella and Gobet 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 treaten in detail. Special emphasize is on numerical experiments: For every ansatz, we include numerical experiments confirming the conjectured accuracy of our methods. Our methods appear to be at least of weak order one and give improved results at the same computational cost compared to algorithms used widely in practice