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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

${y}_{t}={y}_{T}+{\int }_{t}^{T}g\left(s,{y}_{s},{z}_{s}\right)ds+\left({A}_{T}-{A}_{t}\right)-{\int }_{t}^{T}{z}_{s}d{W}_{s},\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],$

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 60G48 Generalizations of martingales