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Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. (English) Zbl 0953.60059
This paper studies the backward stochastic differential equation (BSDE) of the form $y_t = y_T + \int_t^T g(s,y_s,z_s) ds + (A_T-A_t)-\int_t^T z_s dW_s,\quad t \in [0,T],$ where $$W$$ is a Brownian motion and $$g$$ is a non-anticipative Lipschitz-continuous function. As usual, a process $$y$$ is called a supersolution of the BSDE if it is of the above form for some adapted right-continuous, increasing process $$A$$ and some predictable, square-integrable process $$z$$. The main result of the paper is a theorem which asserts that, under suitable integrability conditions, the pointwise monotone limit of a sequence of supersolutions is again a supersolution of the BSDE. Moreover, it is shown that the corresponding integrands $$z$$ converge weakly in $$L^2$$ and strongly in each $$L^p$$ with $$p<2$$; the processes $$A$$ converge weakly in $$L^2$$. As an application of this result, the author proves a generalization of the classical Doob-Meyer decomposition to so-called $$g$$-supermartingales. These processes refer to a suitably defined nonlinear expectation operator in essentially the same way as usual supermartingales to usual expectations. As a second application, the author shows that there exists a minimal supersolution of the BSDE which is subject to fairly general state- and time-dependent constraints.

##### MSC:
 60H99 Stochastic analysis 60H30 Applications of stochastic analysis (to PDEs, etc.) 60G48 Generalizations of martingales
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