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Minimal Errors for Strong and Weak Approximation of Stochastic Differential Equations

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Monte Carlo and Quasi-Monte Carlo Methods 2006

Summary

We present a survey of results on minimal errors and optimality of algorithms for strong and weak approximation of systems of stochastic differential equations. For strong approximation, emphasis lies on the analysis of algorithms that are based on point evaluations of the driving Brownian motion and on the impact of non-commutativity, if present. Furthermore, we relate strong approximation to weighted integration and reconstruction of Brownian motion, and we demonstrate that the analysis of minimal errors leads to new algorithms that perform asymptotically optimal. In particular, these algorithms use a path-dependent step-size control. for weak approximation we consider the problem of computing the expected value of a functional of the solution, and we concentrate on recent results for a worst-case analysis either with respect to the functional or with respect to the coefficients of the system. Moreover, we relate weak approximation problems to average Kolmogorov widths and quantization numbers as well as to high-dimensional tensor product problems.

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Authors and Affiliations

  1. Fakultät für Mathematik und Informatik, Fern Universität Hagen, Lützowstraße 125, 58084, Hagen, Germany

    Thomas Müller-Gronbach

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstraße 7, 64289, Darmstadt, Germany

    Klaus Ritter

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  1. Thomas Müller-Gronbach
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  2. Klaus Ritter
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Editors and Affiliations

  1. Department of Computer Science, Ulm University, Albert-Einstein-Allee 11, 89069, Ulm, Germany

    Alexander Keller

  2. Department of Computer Science, University of Kaiserslautern, 67653, Kaiserslautern, Germany

    Stefan Heinrich

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

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Müller-Gronbach, T., Ritter, K. (2008). Minimal Errors for Strong and Weak Approximation of Stochastic Differential Equations. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_4

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