  We study the approximation of expectations $\operatorname{E}(f(X))$ for
solutions $X$ of stochastic differential equations and functionals $f$ on the
path space by means of Monte Carlo algorithms that only use random bits instead
of random numbers. We construct an adaptive random bit multilevel algorithm,
which is based on the Euler scheme, the L\'evy-Ciesielski representation of the
Brownian motion, and asymptotically optimal random bit approximations of the
standard normal distribution. We numerically compare this algorithm with the
adaptive classical multilevel Euler algorithm for a geometric Brownian motion,
an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process.
