The randomized unbiased estimators of Rhee and Glynn (Operations Research:63(5), 1026-1043, 2015) can be highly efficient at approximating expectations of path functionals associated with stochastic differential equations (SDEs). However, there is a lack of algorithms for calculating the optimal distributions with an infinite horizon. In this article, based on the method of Cui et.al. (Operations Research Letters: 477-484, 2021), we prove that, under mild assumptions, there is a simple representation of the optimal distributions. Then, we develop an adaptive algorithm to compute the optimal distributions with an infinite horizon, which requires only a small amount of computational time in prior estimation. Finally, we provide numerical results to illustrate the efficiency of our adaptive algorithm.
翻译：Rhee 和 Glynn（运筹学：63（5），1026-1043，2015）的随机无偏估计器可以高效地逼近与随机微分方程（SDE）相关的路径函数期望。然而，在具有无限时间范围时，缺乏用于计算最优分布的算法。在本文中，我们基于 Cui 等人的方法（运筹学通讯：477-484，2021），证明了，在温和假设下，最优分布的表示是简单的。然后，我们开发了一种自适应算法，用于计算具有无限时间范围的最优分布，仅需要对先前估计的少量计算时间。最后，我们提供数值结果，以说明自适应算法的效率。
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