Multi-scale methods for stochastic differential equations

Abstract: Standard Monte Carlo methods are used extensively to solve stochastic differential equations. This thesis investigates a Monte Carlo (MC) method called multilevel Monte Carlo that solves the equations on several grids, each with a specific number of grid points. The multilevel MC reduces the computational cost compared to standard MC. When using a fixed computational cost the variance can be reduced by using the multilevel method compared to the standard one. Discretization and statistical error calculations are also being conducted and the possibility to evaluate the errors coupled with the multilevel MC creates a powerful numerical tool for calculating equations numerically. By using the multilevel MC method together with the error calculations it is possible to efficiently determine how to spend an extended computational budget.