We consider the problem of estimating a nested structure of two expectations taking the form $U_0 = E[\max\{U_1(Y), \pi(Y)\}]$, where $U_1(Y) = E[X\ |\ Y]$. Terms of this form arise in financial risk estimation and option pricing. When $U_1(Y)$ requires approximation, but exact samples of $X$ and $Y$ are available, an antithetic multilevel Monte Carlo (MLMC) approach has been well-studied in the literature. Under general conditions, the antithetic MLMC estimator obtains a root mean squared error $\varepsilon$ with order $\varepsilon^{-2}$ cost. If, additionally, $X$ and $Y$ require approximate sampling, careful balancing of the various aspects of approximation is required to avoid a significant computational burden. Under strong convergence criteria on approximations to $X$ and $Y$, randomised multilevel Monte Carlo techniques can be used to construct unbiased Monte Carlo estimates of $U_1$, which can be paired with an antithetic MLMC estimate of $U_0$ to recover order $\varepsilon^{-2}$ computational cost. In this work, we instead consider biased multilevel approximations of $U_1(Y)$, which require less strict assumptions on the approximate samples of $X$. Extensions to the method consider an approximate and antithetic sampling of $Y$. Analysis shows the resulting estimator has order $\varepsilon^{-2}$ asymptotic cost under the conditions required by randomised MLMC and order $\varepsilon^{-2}|\log\varepsilon|^3$ cost under more general assumptions.

翻译：本文考虑一类嵌套期望结构的估计问题，其形式为 $U_0 = E[\max\{U_1(Y), \pi(Y)\}]$，其中 $U_1(Y) = E[X\ |\ Y]$。此类项在金融风险估计与期权定价中广泛出现。当 $U_1(Y)$ 需借助近似计算，但 $X$ 和 $Y$ 的精确样本可得时，已有文献对基于对偶采样的多层蒙特卡罗（MLMC）方法进行了深入研究。在一般条件下，该对偶MLMC估计量能以 $\varepsilon^{-2}$ 量级的计算代价达到均方根误差 $\varepsilon$。若 $X$ 和 $Y$ 亦需近似采样，则需谨慎平衡各近似环节以避免显著计算负担。基于 $X$ 和 $Y$ 近似值的强收敛准则，可采用随机化多层蒙特卡罗技术构建 $U_1$ 的无偏估计，并与 $U_0$ 的对偶MLMC估计相结合，恢复 $\varepsilon^{-2}$ 量级的计算代价。本文另辟蹊径，考虑 $U_1(Y)$ 的有偏多层近似，该方法对 $X$ 的近似样本要求更宽松。进一步扩展考虑 $Y$ 的近似与对偶采样。分析表明：在随机化MLMC所需条件下，所得估计量具有 $\varepsilon^{-2}$ 渐近计算代价；在更一般假设下，代价为 $\varepsilon^{-2}|\log\varepsilon|^3$ 量级。