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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 + t T g(s,y s ,z s )ds+(A T -A t )- t T z s dW s ,t[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:
60H99Stochastic analysis
60H30Applications of stochastic analysis
60G48Generalizations of martingales