Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and Complexity

We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations E f (X T ) of a diffusion (X t ) t ∈[0, T ] when the weak time discretization error induced by the Euler scheme admits an expansion at an order R ≥ 2. The complexity of the estimator grows as R 2 (instead of 2 R in the classical method) and its variance is asymptotically controlled by considering some consistent Brownian increments in the underlying Euler schemes. Some Monte Carlo simulations were carried with path-dependent options (lookback, barrier) 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.