Monte Carlo methods are powerful means to addressing otherwise intractable computational problems. In many settings, however, there may be no known algorithm for generating the exact random samples required to proceed with traditional Monte Carlo methods. In such settings, one typically has to resort to the approximations instead of the exact samples. The errors from such approximations often lead to slower convergence rates, and the biases from the approximation errors are difficult to estimate. In this talk, we will present a simple yet effective and broadly applicable idea for constructing unbiased estimators (i.e., perfect estimation) based only on approximations (i.e., imperfect samples), so as to eliminate such complications and recover the desirable properties of Monte Carlo methods. We will then illustrate the application of the idea in the contexts of stochastic differential equations, equilibrium computation for Markov chains, and rare event simulation of stochastic recurrence equations. We will also discuss its close connection with the standard (biased) multilevel Monte Carlo methods.

Based on the joint works with Bohan Chen, Bert Zwart, and Peter Glynn.