We consider a Markov jump process on a general state space to which we apply
    a time-dependent weak perturbation over a finite time interval. By
    martingale-based stochastic calculus, under a suitable exponential moment bound
    for the perturbation we show that the perturbed process does not explode almost
    surely and we study the linear response (LR) of observables and additive
    functionals. When the unperturbed process is stationary, the above LR formulas
    become computable in terms of the steady state two-time correlation function
    and of the stationary distribution. Applications are discussed for birth and
    death processes, random walks in a confining potential, random walks in a
    random conductance field. We then move to a Markov jump process on a finite
    state space and investigate the LR of observables and additive functionals in
    the oscillatory steady state (hence, over an infinite time horizon), when the
    perturbation is time-periodic. As an application we provide a formula for the
    complex mobility matrix of a random walk on a discrete $d$-dimensional torus,
    with possibly heterogeneous jump rates.

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