The Least-Squares Monte Carlo (LSMC) method has gained popularity in recent years due to its ability to handle multi-dimensional stochastic control problems, including problems with state variables affected by control. However, when applied to the stochastic control problems in the multi-period expected utility models, such as finding optimal decisions in life-cycle expected utility models, the regression fit tends to contain errors which accumulate over time and typically blow up the numerical solution. In this paper we propose to transform the value function of the problems to improve the regression fit, and then using either the smearing estimate or smearing estimate with controlled heteroskedasticity to avoid the re-transformation bias in the estimates of the conditional expectations calculated in the LSMC algorithm. We also present and utilise recent improvements in the LSMC algorithms such as control randomisation with policy iteration to avoid accumulation of regression errors over time. Presented numerical examples demonstrate that transformation method leads to an accurate solution. In addition, in the forward simulation stage of the control randomisation algorithm, we propose a re-sampling of the state and control variables in their full domain at each time t and then simulating corresponding state variable at \(t+1\), to improve the exploration of the state space that also appears to be critical to obtain a stable and accurate solution for the expected utility models.

最小二乘蒙特卡洛 (LSMC) 方法由于能够处理多维随机控制问题，包括受控制影响的状态变量问题，因此近年来受到欢迎。然而，当应用于多周期预期效用模型中的随机控制问题时，例如在生命周期预期效用模型中寻找最优决策，回归拟合往往包含随着时间累积的误差，通常会破坏数值解。在本文中，我们建议转换问题的价值函数以改进回归拟合，然后使用带有受控异方差的拖尾估计或拖尾估计避免在 LSMC 算法中计算的条件期望估计中的重新转换偏差。我们还展示并利用了 LSMC 算法的最新改进，例如通过策略迭代控制随机化，以避免随着时间的推移回归误差的积累。给出的数值例子表明变换方法可以得到准确的解。此外，在控制随机化算法的前向模拟阶段，我们建议在每个时间t对其全域中的状态和控制变量重新采样，然后在\(t+1\)处模拟相应的状态变量，以改进对状态空间的探索，这对于获得预期实用新型的稳定和准确的解决方案也很关键。