Estimating the value of a discounted reward process

This paper provides a differential equation which relates the expected total discounted reward of a reward process to the expected total undiscounted reward of a process which terminates at a negative binomial stopping time. The solution of this equation provides the basis for unbiased estimators of the expected total discounted reward and its derivative with respect to the discount rate. We compare this estimator to other estimators and discuss when it might be more efficient. When rewards are positive we show that the estimator is monotone in the sampled variate.