Existence and uniqueness result for a backward stochastic differential equation whose generator is Lipschitz continuous in y and uniformly continuous in z

摘要

In this note, we extend the result in Lepetier and San Martin (Stat. Probab. Lett. 32:425–430, 1997) by eliminating the condition that (g(t,0,0)) _t∈[0,T] is a bounded process. Furthermore, we prove that if g is Lipschitz continuous in y and uniformly continuous in z, and (g(t,0,0)) _t∈[0,T] is square integrable, then for each square integrable terminal condition ξ, there exists a unique square integrable adapted solution to the one-dimensional backward stochastic differential equation (BSDE) with the generator g, which generalizes the corresponding (one-dimensional) results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008) and Jia (Stat. Probab. Lett. 79:436–441, 2009).