The project is concerned with the quadrature of stochastic differential equations (SDEs), i.e., the computation of expectations E[f(X)] for the solution process X of the SDE. It has applications in various fields such as finance, where the expectation is the risk neutral price of an option. We employ the concept of approximation of probability distributions as the basis for quadrature of SDEs, both, for constructing new deterministic and randomized algorithms, as well as for establishing optimality results. The project consists of three branches
(A) Constructive Quantization of SDEs
(B) Quantization and Random Sampling of Multiple Itô-integrals
(C) Nonlinear Approximation for Lévy-driven SDEs
In branch (A), quantization is employed for the construction of quadrature formulas for SDEs. Furthermore, we investigate deterministic and randomized multi-level algorithms for integration on the sequence space R^N, which is motivated, e.g., by series expansions of stochastic processes. Branch (B) develops quantization based sampling techniques that are used in higher-order Itô-Taylor schemes. These are employed in multilevel Monte Carlo methods for the quadrature of SDEs. In branch (C), we study nonlinear approximation of Lévy processes and construct efficient multilevel Monte Carlo methods for quadrature of Lévy-driven SDEs.
