Some recent improvements to importance splitting∗

The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10, naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but this method requires enough knowledge about the system to decide where to put the levels at hand and select the branching rates. In this paper, we propose to improve the importance splitting algorithm by using adaptive branching rates.This variant of the algorithm may be describe by the formalism of Feynman-Kac Formula, providing a very elegant way of studying its properties. We also propose a second variant using adaptive splitting levels, for which only partial convergence results have been obtained. We will give some numerical examples, including the computation large self avoiding random walks on the 2D lattice.