BSDEs With Integrable Parameters And Nonlinear Expectation

We study the following type of backward stochastic differential equations (BSDEs in short)The random function g is generally called a coefficient (also called the generator), and the random variable ? a terminal condition. A solution is a couple of process(?)s {(yt. zt),0 = t=T} adapted to the filtration (Ft)o=t=T, which have some intcgrability properties depending on the framework imposed by the type of assumptions on g. We often rewrite {(yt, zt), 0 = t= T} as (yt, zt) in short.Let us mention that, under the Lipschitz and square integrable conditions, Par-doux and Peng (1990) gave the first existence and uniqueness results by using both It(?)'s representation theorem and a successive approximation procedure. Since then, many papers have been devoted to existence and (/or) uniqueness results under weaker assumptions, i.e. relaxing the Lipschitz hypothesis on the coefficient. We point out emphatically that all the above work was under the assumption that the terminal condition ? is square integrable. In 1997, El Karoui et al. considered the BSDEs with ? ?L~p(O, Ft, P), where p ? (1,2]. They also get the existence and uniqueness results by using the fixed point theorem. We considered the case of p = 1, and prove the existence and uniqueness result. Peng (1997) introduced the notion of g-martingales by the solutions to BSDEs. When g = 0, nonlinear g-martingales is linear martingales...