Variance reduction methods using quasi-Monte Carlo

Specialeforsvar: Rasmus Hedegaard Kristiansen

Titel: Variance reduction methods using quasi-Monte Carlo

Resume: The topic of the thesis is variance reduction through quasi-Monte Carlo methods. We give a brief overview of the usual Monte Carlo methods, and discuss how to replace them with quasi-Monte Carlo, which are methods based on a set of deterministic points in [0, 1)^d that are chosen to be highly uniform, where d is the dimension of the problem. General theory of such highly uniform point-sets will be discussed, and explicit constructions of some well-known point-sets will be given. We discuss two main topics in the theory of variance reduction in the setting of rare events. One is multi-level splitting, which is a variance reduction technique useful in a Markov chain setup, and which works by partitioning the given problem into a set of easier sub-problems. The other is importance sampling, which has a wide area of applications, and which always works by changing the governing probability measure to one that is more favorable to the random variable or event that we wish to estimate the mean value or probability of. Both multi-level splitting and importance sampling will be fused with quasi- Monte Carlo methods, and we will end up giving numerical examples of these hybrid variance reduction
methods. In particular, multi-level splitting will be applied to an Ornstein-Uhlenbeck process, and importance sampling will be applied to a ruin problem. We compare the different approaches to see if quasi- Monte Carlo offers an improvement.

Vejleder: Jeffrey F. Collamore
Censor: Mette Havning