Håkon Hoel: Weak approximation of stochastic differential equations by a multilevel Monte-Carle method using strong adaptive numerical integration

In this talk we present a multilevel Monte Carlo (MLMC) method for weak approximation of stochastic differential equations (SDE) for which SDE realizations on all levels are generated using an a posteriori adaptive Euler-Maruyama step-size control for the strong error. Strong error adaptivity turns out to be useful for weak approximation MLMC methods since controlling the strong error of realizations at all levels provides a reliable and efficient way to control the statistical error of the weak approximation MLMC estimator. For a large set of low-regularity weak approximation problems the developed adaptive method produces output whose weak error is bounded by $O(\epsilon)$ at the cost $O(\epsilon^{-2}|\log(\epsilon)|^3)$, which is a lower asymptotic cost than typically can be obtained by the uniform time-step MLMC method on the given set of problems. Furthermore, numerical studies are provided for 1D and higher dimensional low-regularity weak approximation problems where the adaptive MLMC method outperforms the uniform time-step MLMC method in terms of computer runtime.