One-dimensional BSDEs with left-continuous, lower semi-continuous and linear-growth generators

This paper deals with a one-dimensional backward stochastic differential equation (BSDE) whose generator gg is of linear growth in (y,z)(y,z), left-continuous and lower semi-continuous (maybe discontinuous) in yy, and continuous in zz. We establish, in this setting, the existence of the minimal solution to the BSDE. And we also prove a comparison theorem and a Levi type theorem for the minimal solutions. They generalize some known results.