2023/01/10 by Yasa Syed, Syed, Yasa, Guanyang Wang +1 · 1 citation
Decision Sciences · Mathematics · #Computation (stat.CO) #Computational Finance (q-fin.CP) #FOS: Computer and information sciences #FOS: Economics and business #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Probability and Risk Models #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2301.04095
openalex publication_date 2023/01/10 · openalex created_date 2023/01/12 · openalex updated_date 2026/07/28
The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer D for the total number of nestings. Standard Monte Carlo methods typically cost at least O(ε-(2+D)) and sometimes O(ε-2(1+D)) to obtain an estimator up to ε-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for D = 1. In this paper, we propose a novel Monte Carlo estimator called READ, which stands for "Recursive Estimator for Arbitrary Depth.'' Our estimator has an optimal computational cost of O(ε-2) for every fixed D under suitable assumptions, and a nearly optimal computational cost of O(ε-2(1 + δ)) for any 0 < δ< \frac12 under much more general assumptions. Our estimator is also unbiased, which makes it easy to parallelize. The key ingredients in our construction are an observation of the problem's recursive structure and the recursive use of the randomized multilevel Monte Carlo method.