Some uniqueness results for one-dimensional BSDEs with uniformly continuous coefficients

In this note we discuss one-dimensional backward stochastic differential equations (BSDEs) with coefficient gg which is uniformly continuous in (y,z)(y,z). As we know, the solution to this kind of BSDE may be non-unique. We prove that, the set of real numbers cc such that the solution of perturbed BSDE with coefficient g+cg+c is non-unique, is at most countable, and we give some necessary and sufficient conditions for the uniqueness for solution to this kind of BSDEs. More importantly, we prove that if gg is independent of yy, the solution of corresponding BSDE is unique.