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（Operations Research:63(5), 1026-1043, 2015）提出的随机无偏估计量在近似随机微分方程（SDEs）路径泛函期望方面具有高效性。然而，目前缺乏计算无限时域最优分布的算法。本文基于Cui等人（Operations Research Letters: 477-484, 2021）的方法，证明在温和假设条件下，最优分布存在简洁表示。进而，我们开发了一种自适应算法，用于计算无限时域最优分布，该算法在先验估计中仅需少量计算时间。最后，通过数值结果验证了所提自适应算法的有效性。