We study the approximation of expectations $E\left(f\left(X\right)\right)$ for solutions $X$ of SDEs and functionals $f:C([0,1],{\mathbb{R}}^r)\to \mathbb{R}$ 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.

我们研究期望值的近似值 $Ë（F（X））$ 解决方案 $X$ SDE和功能 $F：C（[0，\mathrm{1个}]，{\mathbb{[R}}^{\mathrm{[R}}）\to \mathbb{[R}$通过受限的蒙特卡洛算法，该算法可能仅使用随机位而不是随机数。我们考虑功能的最坏情况设置$F$来自最高规范的Lipschitz阶级。我们构造了一个随机位多级Euler算法，并为其误差和成本建立了上限。此外，我们导出了对数因子的匹配下限，该下限对所有随机位蒙特卡洛算法均有效，并且我们证明，对于给定的正交问题，随机位蒙特卡洛算法至少具有与通用随机算法几乎一样的强大功能算法。