The problem of computing the conditional expectation E[f (Y)|X] with least-square Monte-Carlo is of general importance and has been widely studied. To solve this problem, it is usually assumed that one has as many samples of Y as of X. However, when samples are generated by computer simulation and the conditional law of Y given X can be simulated, it may be relevant to sample K $\in$ N values of Y for each sample of X. The present work determines the optimal value of K for a given computational budget, as well as a way to estimate it. The main take away message is that the computational gain can be all the more important that the computational cost of sampling Y given X is small with respect to the computational cost of sampling X. Numerical illustrations on the optimal choice of K and on the computational gain are given on different examples including one inspired by risk management.

翻译：计算条件期望 E[f(Y)|X] 的最小二乘蒙特卡洛方法具有普遍重要性，并得到了广泛研究。为解决该问题，通常假设 Y 与 X 的样本量相同。然而，当样本通过计算机模拟生成且给定 X 时 Y 的条件分布可被模拟时，对每个 X 样本生成 K∈N 个 Y 样本可能更为合理。本文确定了给定计算预算下 K 的最优取值及其估算方法。主要结论是：当给定 X 条件下生成 Y 样本的计算成本相对于生成 X 样本的计算成本越小时，计算效率的提升就越显著。本文通过不同案例（包括一个受风险管理启发的案例）展示了 K 的最优选择及计算增益的数值结果。