Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type.

*(English)*Zbl 0953.60059This 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.

Reviewer: Peter Bank (Berlin)

##### MSC:

60H99 | Stochastic analysis |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60G48 | Generalizations of martingales |