To solve problems using simulation methods, like Monte Carlo (MC) or quasi-Monte Carlo (QMC) methods, it often requires to sample paths of stochastic processes which depend on Brownian motions. While different ways of constructing Brownian paths does not make a difference for MC simulation in terms of accuracy, it may have a big impact for QMC methods. A ``good'' choice of constructing a Brownian path can have a positive effect on the convergence rate of the QMC method, but one may have a drawback in terms of the computational costs.
In this talk I will discuss two challenging questions:
- How can a Brownian path be constructed in a fast way?
- Why can a certain construction method have a positive impact on the efficiency of QMC?
Furthermore, I show how Hermite spaces are related to the analysis of QMC simulation with different Brownian path construction methods.