Often a fixed stepsize implementation for solving SDEs is inappropriate; for example, the dynamics of the system may require a small stepsize over only part of the integration, but a fixed stepsize implementation must use this stepsize for the entire integration. It is natural then to want to implement a numerical method in variable stepsize mode. However there are many difficulties in this. It is important to follow a Brownian path that can be repeated if the integration is to be computed using different initial values (for example). The first attempts at a variable stepsize implementation required the construction of a Brownian tree (binary), but this necessarily restricted the stepsize change to either halving or doubling. In this talk I discuss the difficulties that can arise in using a Brownian Tree/variable stepsize implementation, and I will present a methodology that allows a completely flexible change-of-stepsize while maintaining the integrity of the Brownian path.

MINISIMPOSIUM SESSION: 7. Numerical methods for Stochastic equations