Hierarchical Importance Sampling for Estimating Occupation Time for SDE Solutions

2025-09-17

AI Summary

This paper addresses the problem of estimating tail probabilities for the occupation time below a threshold—i.e., fade duration—of stochastic differential equation (SDE)-driven processes, particularly in rare-event regimes where standard Monte Carlo simulation suffers from prohibitively high variance and computational cost. To overcome this, we propose an importance sampling framework grounded in stochastic optimal control: the optimal importance sampling measure is derived by solving an auxiliary Hamilton–Jacobi–Bellman equation to characterize the most likely (optimal) deviation path. We design both single-level and multilevel importance sampling estimators and, for the first time, establish necessary and sufficient conditions under which the multilevel estimator strictly outperforms its single-level counterpart. Furthermore, we introduce a co-likelihood ratio formulation that ensures overall computational complexity with better-than-second-order convergence. Numerical experiments—including fade-duration estimation in wireless fading channels—demonstrate substantial improvements in both accuracy and efficiency, significantly reducing estimation variance and computational overhead for rare-event simulation.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web data generation and simulation
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

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.

Problem

• Estimating rare event occupation time for SDE solutions
• Developing optimal importance sampling with HJB-PDE control
• Extending single-level to multilevel IS for efficiency

Innovation

• Optimal single level IS estimator using HJB-PDE
• Multilevel IS estimator with enhanced efficiency
• Common likelihood formulation satisfying performance condition