Random bit multilevel algorithms for stochastic differential equations

被引：5

作者：

Giles, Michael B. ^{[1
]}

Hefter, Mario ^{[2
]}

Mayer, Lukas ^{[2
]}

Ritter, Klaus ^{[2
]}

机构：

[1] Univ Oxford, Math Inst, Oxford 0X2 6GG, England

[2] Tech Univ Kaiserslautern, Fachbereich Math, Postfach 3049, D-67653 Kaiserslautern, Germany

来源：

JOURNAL OF COMPLEXITY | 2019年 / 54卷

基金：

英国工程与自然科学研究理事会;

关键词：

Random bits; Multilevel Monte Carlo algorithms; Stochastic differential equations; RESTRICTED MONTE-CARLO; APPROXIMATION; INTEGRATION; ERROR;

D O I：

10.1016/j.jco.2019.01.002

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

We study the approximation of expectations E(f(X)) for solutions X of SDEs and functionals f: C([0, 1], R-r) -> R by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals f from the Lipschitz class w.r.t. the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms. (C) 2019 Elsevier Inc. All rights reserved.

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页数：21

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