Measure-Valued Differentiation for Stationary Markov Chains

We study general state-space Markov chains that depend on a parameter, say, θ. Sufficient conditions are established for the stationary performance of such a Markov chain to be differentiable with respect to θ. Specifically, we study the case of unbounded performance functions and thereby extend the result on weak differentiability of stationary distributions of Markov chains to unbounded mappings. First, a closed-form formula for the derivative of the stationary performance of a general state-space Markov chain is given using an operator-theoretic approach. In a second step, we translate the derivative formula into unbiased gradient estimators. Specifically, we establish phantom-type estimators and score function estimators. We illustrate our results with examples from queueing theory.