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.


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