Variational approach for the adapted solution of the general backward stochastic differential equations under the Bihari condition

In this paper, we study the existence and uniqueness of the adapted solution of a backward stochastic differential equation with a general diffusion coefficient. By using the idea of Brownian bridge, and changing the control term from the diffusion coefficient to the drift coefficient, we prove the existence of the solution under the Bihari condition, which extends the E-well posed condition (Peng, 1994). The uniqueness properties of the solution are also discussed in this paper.