This study considers the estimation of the complementary cumulative distribution function of the occupation time (i.e., the time spent below a threshold) for a process governed by a stochastic differential equation. The focus is on the right tail, where the underlying event becomes rare, and using variance reduction techniques is essential to obtain computationally efficient estimates. Building on recent developments that relate importance sampling (IS) to stochastic optimal control, this work develops an optimal single level IS (SLIS) estimator based on the solution of an auxiliary Hamilton Jacobi Bellman (HJB) partial differential equation (PDE). The cost of solving the HJB-PDE is incorporated into the total computational work, and an optimized trade off between preprocessing and sampling is proposed to minimize the overall cost. The SLIS approach is extended to the multilevel setting to enhance efficiency, yielding a multilevel IS (MLIS) estimator. A necessary and sufficient condition under which the MLIS method outperforms the SLIS method is established, and a common likelihood MLIS formulation is introduced that satisfies this condition under appropriate regularity assumptions. The classical multilevel Monte Carlo complexity theory can be extended to accommodate settings where the single-level variance varies with the discretization level. As a special case, the variance-decay behavior observed in the IS framework stems from the zero variance property of the optimal control. Notably, the total work complexity of MLIS can be better than an order of two. Numerical experiments in the context of fade duration estimation demonstrate the benefits of the proposed approach and validate these theoretical results.

翻译：本研究探讨了随机微分方程控制过程中占据时间（即低于某一阈值的时间）互补累积分布函数的估计问题。重点研究右尾区域，其中基础事件变得罕见，采用方差缩减技术对于获得计算高效的估计至关重要。基于近期将重要性采样与随机最优控制相联系的研究进展，本文通过求解辅助的Hamilton Jacobi Bellman偏微分方程，构建了最优单层重要性采样估计器。求解HJB-PDE的计算成本被纳入总计算工作量，并提出预处理与采样之间的优化权衡以最小化总体成本。将单层重要性采样方法扩展至多层设置以提升效率，从而得到多层重要性采样估计器。建立了多层重要性采样方法优于单层重要性采样方法的充分必要条件，并引入了在适当正则性假设下满足该条件的通用似然多层重要性采样公式。经典的多层蒙特卡洛复杂度理论可扩展至适应单层方差随离散化层级变化的情形。特别地，重要性采样框架中观察到的方差衰减行为源于最优控制的零方差特性。值得注意的是，多层重要性采样的总工作量复杂度可优于二阶。在衰落持续时间估计背景下的数值实验证明了所提方法的优势，并验证了这些理论结果。