2017/01/16 by Fan, ShengJun
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1701.04151
This paper aims at solving a one-dimensional backward stochastic differential equation (BSDE for short) with only integrable parameters. We first establish the existence of a minimal L1 solution for the BSDE when the generator g is stronger continuous in (y,z) and monotonic in y as well as it has a general growth in y and a sublinear growth in z. Particularly, the g may be not uniformly continuous in z. Then, we put forward and prove a comparison theorem and a Levi type theorem on the minimal L1 solutions. A Lebesgue type theorem on L1 solutions is also obtained. Furthermore, we investigate the same problem in the case that g may be discontinuous in y. Finally, we prove a general comparison theorem on L1 solutions when g is weakly monotonic in y and uniformly continuous in z as well as it has a stronger sublinear growth in z. As a byproduct, we also obtain a general existence and unique theorem on L1 solutions. Our results extend some known works.