Existence of solutions to one-dimensional BSDEs with semi-linear growth and general growth generators

This paper is devoted to solving a one-dimensional backward stochastic differential equation (BSDE in short) when the generator gg has a semi-linear growth and a general growth in (y,z)(y,z). This condition is not only strictly weaker than the linear growth condition of gg in (y,z)(y,z), but also the (weak) monotonicity and general growth condition of gg in yy together with the linear growth condition of gg in zz. We establish, in this setting, three existence results on a solution and the minimal (maximal) solution to the BSDE, where the generator gg may be discontinuous in yy. These results virtually unify and improve some existing results.