The Full Bayesian Significance Test (FBST) is adjusted for jump detection in a diffusion process. Under a natural parameterization, pure diffusion can be seen as a precise hypothesis. The evidence measure defined by FBST deals with absolutely continuous posterior distributions, when posterior rates for precise hypotheses are not appropriate. Applications to simulated and real data are shown.38812621271Fondecyt [1070918]PBCT/ACT [13]Laboratorio ANESTOCFondecyt [1070918]PBCT/ACT [13]