A Kind Of Extended Backward Stochastic Differential Equations

The general theory of backward stochastic differential equations(BSDEs) has been de-veloped only recently, beginning in the early 1990s. In a general form, BSDEs were intro-duced by Pardoux and Peng, who proved existence and uniqueness of the solution adaptedthe Brownian filtration under Lipschitz conditions. In the following years, the theory of BS-DEs has been found further important applications and has bocome a powerful tool in-many fields, above all financial mathematics, optimal control and stochastic games,partialdifferential equations, recursion effectiveness, differential effectiveness, option pricing andso on. Because of the so many important applications of BSDEs, it attaracts the interestsof a lot of mathematicans. After many scholars have made the unremitting effort in thisaspect, the theory of BSDEs obtained a rapid development in recent years.The classical Backward Stochastic Differential Equation theory is take the Brown mo-tion as the noise source,and its soiution should be adapted to the filtration generated bythe driving Brownian motion. But the Brown motion is one kind of extreme idealizedmodel, causes the classical Backward Stochastic Differential Equation theory to receivecertain limit in the application. From the beginning, many authors attempted to improvethe theory of the classical BSDEs and a lot of good results have been achieved.In this paper, we introduce the follow type of extended BSDEs Y_t=ξ+integral from n=t to T (g(s,Y_s,M_s)ds-(M_T-M_t))where theξis p(1＜p≤+∞)-integrable, M is a L~p-integrable martingle. Here, we firstgive the definition of solution of the above equation,then under the Lipschitz conditionwe dicuss the existence and uniqueness of the solution,the comparison of the solutionand a kind of the extended g-expectation and g-martingale. We obtain the following threeresults related to the above extended BSDEs: the first is the theorem of existence anduniqueness of the solution of the extended BSDEs, the second is the comparison theoremof the extended BSDEs, and the last is the notion of the extended g-expectation andg-martingale and some properties of them.