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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.

Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states