The dependence of the yield of high-order harmonic generation (HHG) on several important experimental parameters
has been successfully modeled in the last 20 years by taking into account the single atom response and propagation
effects. We extended this description by adding a stimulated emission process and named it x-ray parametric
amplification (XPA). Beyond the super-quadratic increase of the XUV signal, which can be explained only in a limited
pressure range by HHG theory, other observed characteristics like exponential growth, gain narrowing, strong blue-shift,
beam divergence, etc. and their dependence on laser intensity and gas pressure can be explained accurately only by the
new XPA model. We experimentally demonstrated XPA in Argon in the spectral range of 40-50 eV in excellent
agreement with the theory. XPA holds the promise to realize a new class of bright x-ray sources for spectroscopy.