In this paper a novel approach is proposed to address the problem of deriving non-stationary stochastic processes which are compatible in the mean sense with a given (target) response (uniform hazard) spectrum (UHS) as commonly desired in the aseismic structural design regulated by contemporary codes of practice. The appealing feature of the approach is that it is non-iterative and "one-step". This is accomplished by solving a standard over-determined minimization problem in conjunction with appropriate median peak factors. These factors are determined by a plethora of reported new Monte Carlo studies which on their own possess considerable stochastic dynamics merit. In the proposed approach, generation and treatment of samples of the processes individually on a deterministic basis is not required as is the case with the various "two-step" approaches found in the literature addressing the herein considered task. The applicability and usefulness of the approach is demonstrated by furnishing extensive numerical data associated with the elastic design UHS of the current European (EC8) and the Chinese (GB 50011) aseismic code provisions. Purposely, simple and thus attractive from a practical viewpoint, uniformly modulated processes assuming either the Kanai-Tajimi (K-T) or the Clough-Penzien (C-P) spectral form are employed. The Monte Carlo studies yield damping and duration dependent median peak factor spectra, given in a polynomial form, associated with the first passage problem for UHS compatible K-T and C-P uniformly modulated stochastic processes. Hopefully, the herein derived stochastic processes and median peak factor spectra can be used to facilitate the aseismic design of structures regulated by contemporary code provisions in a Monte Carlo simulation-based or stochastic dynamics-based context of analysis.