  We introduce a new class of estimators for the linear response of steady
states of stochastic dynamics. We generalize the likelihood ratio approach and
formulate the linear response as a product of two martingales, hence the name
"martingale product estimators". We present a systematic derivation of the
martingale product estimator, and show how to construct such estimator so its
bias is consistent with the weak order of the numerical scheme that
approximates the underlying stochastic differential equation. Motivated by the
estimation of transport properties in molecular systems, we present a rigorous
numerical analysis of the bias and variance for these new estimators in the
case of Langevin dynamics. We prove that the variance is uniformly bounded in
time and derive a specific form of the estimator for second-order splitting
schemes for Langevin dynamics. For comparison, we also study the bias and
variance of a Green-Kubo estimator, motivated, in part, by its variance growing
linearly in time. Presented analysis shows that the new martingale product
estimators, having uniformly bounded variance in time, offer a competitive
alternative to the traditional Green-Kubo estimator. We compare on illustrative
numerical tests the new estimators with results obtained by the Green-Kubo
method.
