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The randomized unbiased estimators of Rhee and Glynn (OperationsResearch:63(5), 1026-1043, 2015) can be highly efficient at approximatingexpectations of path functionals associated with stochastic differentialequations (SDEs). However, there is a lack of algorithms for calculating theoptimal distributions with an infinite horizon. In this article, based on themethod of Cui et.al. (Operations Research Letters: 477-484, 2021), we provethat, under mild assumptions, there is a simple representation of the optimaldistributions. Then, we develop an adaptive algorithm to compute the optimaldistributions with an infinite horizon, which requires only a small amount ofcomputational time in prior estimation. Finally, we provide numerical resultsto illustrate the efficiency of our adaptive algorithm.

Optimal distributions for randomized unbiased estimators with an infinite horizon and an adaptive algorithm