We consider the problem of approximating the marginal distribution of the solution of a stochastic differential equation (SDE) by probability measures with finite support, i.e., by quadrature formulas with positive weights summing up to one.  We study deterministic algorithms in a worst case analysis with respect to classes of SDEs, which are defined in terms of smoothness constraints for the coefficients of the equation. The worst case error of an algorithm is defined in terms of a metric on the space of probability measures on the state space of the solution. We present and discuss sharp asymptotic bounds on the respective N-th minimal error, which is the smallest error that can be achieved by any such algorithm with computational cost bounded by N. We also provide a sketch of the proof of this result.

 

Thomas Mueller received his Ph.D. in Mathematics at the Free University of Berlin in 1993. And habilitation in Mathematics at the Technical University of Darmstadt in 2001.  Between 2002 and 2008, he held lecturer positions at the University of Bayreuth, the University of Magdeburg and the Technical University of Darmstadt. Since 2008, he is Professor at the University of Passau at the Faculty of Computer Science and Mathematics. His research interests include: Numerical Methods for stochastic (partial) differential equations, Quantization and infinite dimensional quadrature, Nonlinear approximation of stochastic processes, Monte-Carlo algorithms, Complexity of continuous problems.