Stochastic Differential Equations (SDE) are non-deterministic processes used to model natural processes with uncertainty, such as multi-scale particle dynamics and the evolution of financial assets. A practical way of generating approximations of realizations of SDE is by numerical integration on a mesh of uniformly spaced time increments, but in settings with low regularity, one may achieve substantial improvements in the accuracy by adapting the mesh step-size to the instabilities of the considered problem. Monte Carlo (MC) methods are a 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 MC methods which by sampling stochastic realizations on a hierarchy of resolutions may reduce the computational cost of weak approximations by orders of magnitude.
In this talk we give an outline of MLMC and present a recently developed application of MLMC for weak approximations of SDE with a posteriori step-size control in the Euler--Maruyama numerical integration of SDE realizations. A numerical example illustrating the potential efficiency gain of adaptive time stepping MLMC over uniform time stepping MLMC is provided for a low regularity problem, and the challenges and possible limitations of adaptive numerical integration of SDE will be discussed.