Minimal errors for strong and weak approximation of stochastic differential equations.

*(English)*Zbl 1140.65305
Keller, Alexander (ed.) et al., Monte Carlo and quasi-Monte Carlo methods 2006. Selected papers based on the presentations at the 7th international conference ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Ulm, Germany, August 14–18, 2006. Berlin: Springer (ISBN 978-3-540-74495-5/hbk). 53-82 (2008).

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.

For the entire collection see [Zbl 1130.65003].

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.

For the entire collection see [Zbl 1130.65003].

##### MSC:

65C30 | Numerical solutions to stochastic differential and integral equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

34F05 | Ordinary differential equations and systems with randomness |

65L70 | Error bounds for numerical methods for ordinary differential equations |

##### Keywords:

minimal errors; algorithms; systems of stochastic differential equations; strong approximation; Brownian motion; weak approximation; worst-case analysis; average Kolmogorov widths; quantization numbers
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\textit{T. Müller-Gronbach} and \textit{K. Ritter}, in: Monte Carlo and quasi-Monte Carlo methods 2006. Selected papers based on the presentations at the 7th international conference `Monte Carlo and quasi-Monte Carlo methods in scientific computing', Ulm, Germany, August 14--18, 2006. Berlin: Springer. 53--82 (2008; Zbl 1140.65305)

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