  We consider estimating an expected infinite-horizon cumulative discounted
cost/reward contingent on an underlying stochastic process by Monte Carlo
simulation. An unbiased estimator based on truncating the cumulative cost at a
random horizon is proposed. Explicit forms for the optimal distributions of the
random horizon are given, and explicit expressions for the optimal random
truncation level are obtained, leading to a full analysis of the bias-variance
tradeoff when comparing this new class of randomized estimators with
traditional fixed truncation estimators. Moreover, we characterize when the
optimal randomized estimator is preferred over a fixed truncation estimator by
considering the tradeoff between bias and variance. This comparison provides
guidance on when to choose randomized estimators over fixed truncation
estimators in practice. Numerical experiments substantiate the theoretical
results.
