Large Sample Properties of Weighted Monte Carlo Estimators
Large Sample Properties of Weighted Monte Carlo Estimators
复制标题
加权蒙特卡罗估计量的大样本特性
作者：
P. Glasserman;Bin Yu
提高仿真精度的一般方法是使用有关具有已知预期值的辅助控制变量的信息来改进未知量的估计。我们分析了加权蒙特卡洛估计器，通过将权重应用于独立复制来实现这一想法。选择权重来约束控制变量的加权平均值。我们区分两种情况（无偏和有偏），具体取决于控件的加权平均值是否被限制为等于其预期值或其他值。在这两种情况下，约束的数量通常小于复制的数量，因此可能有许多可行的权重。我们通过最小化受控制变量约束的权重的可分离凸函数来选择最大均匀权重。这种形式的估计器在多种环境中出现（有时是隐含的），其中至少有两种情况出现在金融领域：根据市场数据校准模型（如 Avellaneda 等人 2001 年的工作）以及计算条件预期以对美式期权进行定价。随着重复次数的增加，我们分析这些估计器的属性。我们证明，在无偏的情况下，加权蒙特卡罗减少了渐近方差，并且大类中的所有凸目标函数产生的估计量在强意义上彼此非常接近。相反，在有偏差的情况下，目标函数的选择确实很重要。我们明确地展示了目标的选择如何决定估计器收敛的极限。
A general approach to improving simulation accuracy uses information about auxiliary control variables with known expected values to improve the estimation of unknown quantities. We analyze weighted Monte Carlo estimators that implement this idea by applying weights to independent replications. The weights are chosen to constrain the weighted averages of the control variables. We distinguish two cases (unbiased and biased), depending on whether the weighted averages of the controls are constrained to equal their expected values or some other values. In both cases, the number of constraints is usually smaller than the number of replications, so there may be many feasible weights. We select maximally uniform weights by minimizing a separable convex function of the weights subject to the control variable constraints. Estimators of this form arise (sometimes implicitly) in several settings, including at least two in finance: calibrating a model to market data (as in the work of Avellaneda et al. 2001) and calculating conditional expectations to price American options. We analyze properties of these estimators as the number of replications increases. We show that in the unbiased case, weighted Monte Carlo reduces asymptotic variance, and that all convex objective functions within a large class produce estimators that are very close to each other in a strong sense. In contrast, in the biased case the choice of objective function does matter. We show explicitly how the choice of objective determines the limit to which the estimator converges.