  Recent progress has been made with Adaptive Multiple Importance Sampling
(AMIS) methods that show improvement in effective sample size. However,
consistency for the AMIS estimator has only been established in very restricted
cases. Furthermore, the high computational complexity of the re-weighting in
AMIS (called balance heuristic) makes it expensive for applications involving
diffusion processes. In this work we consider sequential and adaptive
importance sampling that is particularly suitable for diffusion processes. We
propose a new discarding-re-weighting scheme that is of lower computational
complexity, and we prove that the resulting AMIS is consistent. Using numerical
experiments, we demonstrate that discarding-re-weighting performs very similar
to the balance heuristic, but at a fraction of the computational cost.
