We characterize the optical response of a three-level atom subjected to an incoherent pump and continuously illuminated with a weak, quasi-resonant probe field. To this end, we apply a wavefunction approach based on QED Hamiltonian perturbation theory which allows for a reduction of the atomic dynamics to that of an effective two-level atom, and for an implementation of the incoherent effects that respects unitarity. Using a diagrammatic representation, we identify and classify all the radiative processes. This allows us to compute the scattered power, the spontaneous emission, and the stimulated emission, as well as the total cross sections of extinction, absorption and scattering. We find that, beside a general enhancement of the linewidth and an attenuation of the spectral amplitudes, the pump reduces the nonradiative losses and provides gains in the form of stimulated emission and incoherent radiation. For sufficiently strong pump, gains and losses compensate, resulting in the vanishing of extinction. In particular, for negligible nonradiative losses, extinction vanishes for a pumping rate of $(1+sqrt{5})/2$ times that of the natural decay.
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