Filtered Monte Carlo

By a filtered Monte Carlo estimator we mean 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 Levy's formula gives rise to another.

We establish variance inequalities complementing compensator identities. Among estimators that are (Stieltjes) stochastic integrals, we show that filtering reduces variance if the integrand and the increments of the integrator have conditional positive correlation. We also provide more primitive hypotheses that ensure this condition, making use of stochastic monotonicity properties. Our most detailed conditions apply in a Markov setting where monotone, up-down, and convex generators play a central role. We give examples. As an application of our results, we compare certain estimators that do and do not exploit the property that Poisson arrivals see time averages.