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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00120898
Contributor : Gilles Pagès <>
Submitted on : Monday, December 17, 2007 - 1:38:45 PM
Last modification on : Monday, December 14, 2020 - 9:43:26 AM
Long-term archiving on: : Friday, September 24, 2010 - 10:23:25 AM

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Gilles Pagès. Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and complexity. Monte Carlo Methods and Applications, De Gruyter, 2007, 13 (1), 37-70 ; http://dx.doi.org/10.1515/MCMA.2007.003. ⟨10.1515/MCMA.2007.003⟩. ⟨hal-00120898v4⟩

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