Existence and uniqueness on $L^1$ solutions of multidimensional BSDEs with generators of stochastic one-sided Osgood type

Abstract: By imposing an additional integrability condition on the first component of the solution, this paper establishes an existence and uniqueness result for $L^1$ solutions of multidimensional backward stochastic differential equations (BSDEs) with a general terminal time when the generator $g$ satisfies a stochastic one-sided Osgood condition along with a general growth condition in the state variable $y$, and a stochastic Lipschitz condition in the state variable $z$, extending and strengthening Theorems 1 and 2 of Fan [J. Theor. Probab. 31(2018)]. Two general stochastic Gronwall-type and Bihari-type inequalities along with some innovative techniques dealing with stochastic coefficients and weaker integrability conditions play crucial roles in our proofs, and can be useful in further study on the adapted solution of BSDEs.