  We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for
weak approximation of stochastic models. The CMLMC algorithm solves the given
approximation problem for a sequence of decreasing tolerances, ending when the
required error tolerance is satisfied. CMLMC assumes discretization hierarchies
that are defined a priori for each level and are geometrically refined across
levels. The actual choice of computational work across levels is based on
parametric models for the average cost per sample and the corresponding weak
and strong errors. These parameters are calibrated using Bayesian estimation,
taking particular notice of the deepest levels of the discretization hierarchy,
where only few realizations are available to produce the estimates. The
resulting CMLMC estimator exhibits a non-trivial splitting between bias and
statistical contributions. We also show the asymptotic normality of the
statistical error in the MLMC estimator and justify in this way our error
estimate that allows prescribing both required accuracy and confidence in the
final result. Numerical results substantiate the above results and illustrate
the corresponding computational savings in examples that are described in terms
of differential equations either driven by random measures or with random
coefficients.
