  We consider numerical schemes for computing the linear response of
steady-state averages of stochastic dynamics with respect to a perturbation of
the drift part of the stochastic differential equation. The schemes are based
on Girsanov's change-of-measure theory to reweight trajectories with factors
derived from a linearization of the Girsanov weights. We investigate both the
discretization error and the finite time approximation error. The designed
numerical schemes are shown to be of bounded variance with respect to the
integration time, which is a desirable feature for long time simulation. We
also show how the discretization error can be improved to second order accuracy
in the time step by modifying the weight process in an appropriate way.
