A Kind Of Backward Stochastic Differential Equations Driven By Continuous Local Martingales

Pardoux and Peng(1990) showed the existence and uniqueness of the adapted solution of backward stochastic differential equations(BSDES) driven by Brown motion under the condition that the coefficients are uniformly Lipschitz.A comparison theorem for solutions of one-dimensional BSDE was established by Peng in 1992. General speaking, Lipschitz condition is strong, some efforts have been done in relaxing the Lipschitz hypothesis on the coefficients. In particular, Lepeltier and Martin(1997)recently obtained the existence of solutions under continuity and linear growth conditions. Liu Jicheng and Ren Jiagang generalized the comparison theorem to the case.The classical BSDE theory is taken the Brown motion as the noise source, but the Brown motion is one kind of extreme idealized model, which causes the classical BSDE theory to receive certain limit in the application. Therefore, many scholars try to study the BSDE driven by other noise. For example, Existence and uniqueness results of the solutions of BSDE with continuous local martingle under Lipschitz condition are obtained, and some properties of the solutions are further discussed by Li Juan(Shan Dong University).This extended the classical BSDE theory. Existence of solutions on BSDE is very useful for the BSDE theory. Moreover, it could be applied to financial theory research.In this paper, we discuss some issues of BSDE with continuous local martingale where the coefficient satisfy continuous and linear growth conditions. We obtain the existence theorem of solutions and comparison theorem.