Running bottom quark mass at the GUT scale

In the framework of the Minimal Supersymmetric Standard Model (MSSM) we establish the relation between the bottom quark mass mb at the electroweak and the GUT scale. We describe in detail the transition from the MS to the DR scheme, the matching between QCD and the MSSM, and the running both in the effective and full theory. Our set-up allows for a consistent analysis of mb(μGUT) up to three loops which significantly reduces the dependence on the matching scale between QCD and the MSSM