On the set of solutions of a BSDE with continuous coefficient

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

“…Since the work of Pardoux and Peng (1990), there are many works attempting to relax the Lipschitz condition for getting the existence and uniqueness of solution, for instance Pardoux and Peng (1992), Lepeltier and Martin (1997), Pardoux (1996), Bahlali (2001), Kobylanski (2000), Lepeltier and Martin (1998), Hamadéne (2003) and Briand and Hu (2008), etc. In the case where g is only continuous and of linear growth in (y, z) and n = 1, Lepeltier and Martin (1997) proved that there is at least one solution, actually there is either one or uncountably many solutions in this situation (see Jia and Peng (2007)). …”

The Equivalence between Uniqueness and Continuous Dependence of Solution for BDSDEs

“…) t∈[0,T ] are the minimal solution and maximal solution of BDSDE (7). Now, we give our result for the general case.…”

“…In particular, many efforts have been made to relax the assumption on the generator. For instance, Lepeltier and San Martin [10] have proved the existence of a solution for the case when the generator is only continuous with linear growth, and Jia and Peng [7] obtained that BSDE has either one or uncountably many solutions, if the generator satisfies the conditions given in [10]. Jia and Yu [8] studied the equivalence between uniqueness and continuous dependence of solution for BSDEs with continuous coefficient.…”

“…Now, we give our result for the general case. (iii) Uniqueness: there exists a unique solution of BDSDE (7) with λ = λ 0 , that is, the solution of (f λ 0 , g λ 0 , T, ξ λ 0 ) is unique.…”

A Kneser-type theorem for backward doubly stochastic differential equations

“…For instance, for the one-dimensional case, Lepeltier and San Martin [11] proved the existence of a solution to BSDEs under the assumption of continuous coefficient. Recently, Jia and Peng [5] studied the structure of the solutions to BSDEs with continuous coefficients.…”