We report the appearance of superconductivity under hydrostatic pressure (0.35 to 2.5GPa) in Sr0.5RE0.5FBiS2 with RE = Ce, Nd, Pr and Sm. The studied compounds, synthesized by solid state reaction route, are crystallized in tetragonal P4/nmm space group. At ambient pressure though the RE = Ce exhibit the onset of superconductivity below 2.5K, the Nd, Pr and Sm samples are not superconducting down to 2K. With application of hydrostatic pressure (up to 2.5GPa), superconducting transition temperature is increased to around 10K for all the studied samples. The magneto-transport measurements are carried out on all the samples with maximum Tc i.e., at under 2.5GPa pressure and their upper critical fields are determined. The new superconducting compounds appear to be quite robust against magnetic field but within Pauli paramagnetic limit. The new superconducting compounds with various RE (Ce, Nd, Pr and Sm) belonging to Sr0.5La0.5FBiS2 family are successfully synthesized for the first time and superconductivity is induced in them under hydrostatic pressure.