Abstract
With the test function method and a localization technique, a scalar backward stochastic differential equation (BSDE for short) subject to an L^1 terminal condition is shown to have an L^1 solution when the generator g(t,y,z) has a one-sided linear growth in y and a logarithmic sub-linear growth in z, which improves some existing results. A new idea to study the existence of an adapted solution to a BSDE is given. When the generator g(t,y,z) additionally has an extended monotonicity in y and a logarithmic uniform continuity in z, we further establish a comparison theorem for the L^1 solutions to the above BSDEs, which yields immediately the uniqueness of the solution. This a joint work with Ying Hu and ShanJian Tang.