Numerical Analysis: Adaptive multilevel Monte Carlo simulation
Stochastic Differential Equations (SDE) are non-deterministic processes used to model natural processes with uncertainty, such as micro scale particle dynamics and the evolution of financial assets. A practical way of generating realizations of SDE is by numerical integration on a mesh of uniformly spaced time increments is, but in settings with low regularity one may achieve substantial improvements in the accuracy by adapting the mesh increments to the drift and diffusion coefficients. Monte Carlo methods is class of convenient and robust algorithms for approximating quantities of interest of a given stochastic model through sample averaging of stochastic realizations. Multilevel Monte Carlo (MLMC) is an extension of classical Monte Carlo methods which by sampling stochastic realizations on hierarchy of different resolutions reduce the computational cost of Monte Carlo approximations; see http://people.maths.ox.ac.uk/gilesm/mlmc_community.html.
In this project we develop an adaptive time-integration MLMC method for weak approximations of quantities of interest of SDE. SDE realizations are sampled by means of numerical time-integration, where the mesh is adaptively refined in order to compute the given quantity of interest to a specified accuracy at a minimal cost. Under some assumptions, we are able to prove that our method is asymptotically accurate and have close to optimal computational complexity.