We present a method for calculating the asymmetric quantum current noise in systems described by a Born-Markov master equation.  The technique is particularly suited to describing mesoscopic devices containing Josephson junctions.   As an example we calculate the full frequency dependent current noise in a superconducting single electron transistor tuned to the double Josephson quasiparticle resonance.  Our results are consistent with those obtained in a recent experiment which was able to probe the asymmetric noise.