A filtered Monte Carlo estimator is one whose constituent parts-summands or integral increments-are conditioned on an increasing family of σ-fields. Unbiased estimators of this type are suggested by compensator identities. Replacing a point-process integrator with its intensity gives rise to one class of examples; exploiting Lévy’s formula gives rise to another. Variance inequalities complementing compensator identities are established. Among estimators that are (Stieltjes) stochastic integrals, it is shown that filtering reduces variance if the integrand and the increments of the integrator have conditional positive correlation. More primitive hypotheses that ensure this condition, making use of stochastic monotonicity properties are also provided. The most detailed conditions apply in a Markov setting where monotone, up-down, and convex generators play a central role. Examples are given. As an application of the present results, the paper compares certain estimators that do and do not exploit the property that Poisson arrivals see time averages.
