统计Romberg外推：一种新的方差缩减方法及其在期权定价中的应用
Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
摘要：
研究了随机微分方程解$\mathbb{E}f(X_T)$的Monte Carlo逼近问题，其中$X$为随机微分方程的解，$f$为给定函数.本文介绍了一种新的方差缩减方法，它可以看作是Romberg外推法的统计模拟。也就是说，我们使用两个欧拉格式与步骤$δ$和$δ^β,0<β<1$。这导致了一种算法，对于给定的统计误差水平，该算法的复杂性明显低于标准蒙特卡罗方法的复杂性。我们分析了一般（可能退化）扩散的背景下，该算法的渐近误差。为了找到最优的$β$（结果是$β=1/2$），我们在Jacod和Protter关于Euler格式误差渐近分布的结果的基础上建立了一个中心极限型定理.我们测试我们的方法在不同的例子。特别是，我们将其调整为亚洲选项。在这种情况下，我们有一个CLT，作为副产品，一个明确的扩展的离散化误差。
英文摘要：
We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo algorithm, where $X$ is the solution of a stochastic differential equation and $f$ is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps $δ$ and $δ^β,0<β<1$. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal $β$ (which turns out to be $β=1/2$), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.
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