Error analysis and optimal sample allocation for randomized unbiased estimation: the single term estimator

Randomized unbiased estimation can be highly efficient for approximating expectations of functionals of stochastic differential equations. However, in practice, the optimal computational efficiency is rarely attained due to errors arising from parameter estimation and truncation. The effects of these errors on computational efficiency have not been investigated. In this article, we study the error analysis and optimal sample allocation for the single term estimator. Specifically, under mild assumptions on the estimated parameters, we develop a unified error analysis that incorporates both sources of error, and establish the relevant convergence rates for the computational efficiency loss. Based on this analysis, we further investigate the optimal allocation of the computational budget for parameter estimation, and propose several efficient sample allocation strategies. Numerical experiments support the theoretical convergence rates and demonstrate the effectiveness of our proposed methods.