Capturing rare events is crucial for accurate risk assessment and its successful management. An example in finance is computing the probability of a large, but rare, loss from a financial portfolio. Approximating expectations involving such rare events is difficult because, when using Monte Carlo, many of the generated samples do not contribute to the final outcome and the expensive samples are effectively wasted. Adaptive sampling methods resolve this issue by spending some minimal computational effort to determine if a sample is of interest and, only if it is, the sample accuracy is then improved by spending further computational effort.  Using concepts from stochastic analysis, probably theory and numerical analysis, this project will look at applying adaptive methods to compute outputs depending on stochastic differential equations rather than simple random variables.