  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.
