Constructive Quantization and Multilevel Algorithms for Quadrature of SDEs

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Different views on quadrature problems for stochastic differential equations (SDEs) lead to rather different algorithmic approaches, e.g., PDE methods based on the Fokker-Planck equation, deterministic or randomized high-dimensional numerical integration for explicitly solvable SDEs, and Monte Carlo simulation of SDEs. In this project 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. Our research is devoted to(I) Constructive Quantization of Systems of SDEs(II) Quadrature of Discontinuous Functionals(III) Multilevel Methods for Lévy-driven SDEs(IV) Multilevel Quadrature on the Sequence Space and Malliavin Regularity(V) A Numerical Toolbox in OpenCL/CUDAPart (I) is concerned with the construction of discrete distributions to approximate a target distribution and its application to the quadrature problem. In (II), Malliavin techniques and numerical schemes of higher order are analyzed for quadrature of discontinuous functionals. In (III), we develop simulation techniques for Lévy processes for the design of multilevel schemes and an analysis for continuous and discontinuous functionals is conducted. Recent progress in high-dimensional integration motivates the study in (IV). We build parallel implementations of the algorithms from (II) and (III) that make full use of the massive parallel processing units hidden in today’s graphic cards.