A uniqueness theorem for solution of BSDEs

where W is a standard d-dimensional Brownian motion on a probability space (Ω,F , (Ft)0≤t≤T , P ) with (Ft)0≤t≤T the filtration generated by W . The function g : Ω × [0, T ] × R × R → R is called generator of (1.1). Here T is the terminal time, and ξ is a R-valued FT -adapted random variable; (g, T, ξ) are the parameters of (1.1). The solution (yt, zt)t∈[0,T ] is a pair of Ft-adapted and square integrable processes. Nonlinear BSDEs were first introduced by Pardoux and Peng [7], who proved the existence and uniqueness of a solution under suitable assumptions on g and ξ, the most standard of which are the Lipschitz continuity of g with respect to (y, z) and the square integrability of ξ. An interesting and important question is to find weaker conditions rather than the Lipschitz one, under which the BSDE (1.1) still has a unique solution. As a matter of fact, there have been several works, such as Pardoux and Peng [8], Kobylanski [4] and Briand-Hu [1], etc. In this note, we will give a new sufficient condition for the uniqueness of the solution to BSDEs. In fact, this problem came from a lecture given by Peng at a seminar of Shandong University on Oct. 2005. In his lecture, Peng conjectured that if g is Hölder continuous in z and independent of y, then (1.1) has a unique solution.