On the Optimal Design of the Randomized Unbiased Monte Carlo Estimators

Abstract

In recent literature, a new class of unbiased Monte Carlo estimators have been proposed, which is based on truncating a telescopic representation of the expectation of a functional of the stochastic process at an independent random level. The generality of the method lies in that it can translate any sequence of asymptotically unbiased estimators into a truly unbiased estimator. However, since any independent truncation level can lead to an unbiased estimator, its optimal choice is crucial for the successful implementation of this unbiased Monte Carlo method in financial engineering applications. We develop a novel efficient algorithm to locate the optimal distribution of the random truncation level in general, which answers positively an open question posted in Rhee and Glynn (2015), and we rigorously establish the convergence and optimality of the algorithm and derive its exact complexity. Our algorithm has a much lower complexity as compared to the literature, and numerical examples confirm our findings. The proposed algorithm is shown to be also applicable to optimization problems arising in contextual areas of supply chain management.