  We propose a multi-step Richardson-Romberg extrapolation method for the
computation of expectations $E f(X_{_T})$ of a diffusion $(X_t)_{t\in [0,T]}$
when the weak time discretization error induced by the Euler scheme admits an
expansion at an order $R\ge 2$. The complexity of the estimator grows as $R^2$
(instead of $2^R$) and its variance is asymptotically controlled by considering
some consistent Brownian increments in the underlying Euler schemes. Some Monte
carlo simulations carried with path-dependent options (lookback, barriers)
which support the conjecture that their weak time discretization error also
admits an expansion (in a different scale). Then an appropriate
Richardson-Romberg extrapolation seems to outperform the Euler scheme with
Brownian bridge.
