Multi-step Richardson-Romberg Extrapolation：方差控制和复杂性的备注
Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and complexity
摘要：
本文提出了一种多步Richardson-Romberg外推方法，用于计算扩散方程$(X_t)_{t\in [0,T]}$的期望$E f(X_{_T})$，该方法考虑了由Euler格式引起的弱时间离散化误差在$R\ge 2$阶上的展开.估计量的复杂性随着$R^2$（而不是$2^R$）的增长而增长，并且通过考虑底层Euler格式中的一些一致布朗增量来渐近控制其方差。一些蒙特卡洛模拟进行路径依赖的选项（回顾，障碍），支持的猜想，他们的弱时间离散化误差也承认一个扩展（在不同的规模）。然后，适当的Richardson-Romberg外推似乎优于带布朗桥的欧拉格式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
