A General Result Of Backward Stochastic Volterra Integral Equation With General Martingale

In this paper we will consider the backward stochastic Volterra integral equations (BSVIEs in short) with general martingale. Backward stochastic integral equation is the natural development of backward stochastic differ-ential equation (BSDE in short). There has been more than 20 years since BSDE was first proposed and has formed a rich and improved system theory which become one of the most popular areas in the research of probability theory and stochastic analysis. Its one of the new development directions is backward stochastic integral equations. Stochastic Volterra integral equations were firstly studied by Berger and Mizel in [18] and [19], then they were inves-tigated by Protter in [15] and Pardoux and Protter in [20]. A kind of nonlinear backward stochastic Volterra integral equations was firstly introduced by Lin in [10], then Yong investigated a more general version of BSVIEs in [5]. But lto formula which is widely used in the research of stochastic analysis cannot be used easily here because of BSVIEs'complicated measurability. So Yong proposed the definition of M-solution in [7], Then Wang and Shi proposed the definition of S-solution and simplified Yong's proof in [17]. In this paper, we have researched the backward stochastic Volterra integral equations with gen-eral martingale on the basis of Wang and Shi's paper and obtain the existence and uniqueness for M-solution, duality principle and comparison theorems of BSVIEs and other results. So the theory of backward stochastic Volterra in-tegral equation is developed. The article is mainly divided into four parts:First of all, backward stochastic Volterra integral equation and some related results is introduced. Then in the part of preliminaries some necessary definitions of space and norm are given. In the third part, existence and uniqueness results of adapted M-solution of BSVIE with general martingale under Lipschitz condition is proved, in terms of both the M-solutions introduced in [7] and the adapted solutions in[10]. The article provide a method of proof which is much briefer than the one in [7] to prove the existence and uniqueness of adapted M-solution of BSVIEs. At last the duality principles of linear BSVIEs driven by general martingale is given which prepared for the proof of the comparison theorem and then the comparison theorem of linear BSVIEs is presented by duality principles.