EXISTENCE AND UNIQUENESS OF MULTIDIMENSIONAL BSDEs AND OF SYSTEMS OF DEGENERATE PDEs WITH SUPERLINEAR GROWTH GENERATOR

In the first part of this paper, we deal with the unique solvability of multidimensional backward stochastic differential equations (BSDEs) with a p-integrable terminal condition (p > 1) and a superlinear growth generator. We introduce a new local condition on the generator (assumption (H.4)) and then show that it ensures the existence and uniqueness, as well as the L-p-stability, of solutions. The assumptions that we impose on the generator are local in the three variables y, z, omega, and therefore we also cover the BSDEs with stochastic Lipschitz coefficient. Our conditions on the generator go beyond all existing ones in the literature. For instance, the generator is not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. Furthermore, it could be neither locally monotone in the y-variable nor locally Lipschitz in the z-variable. Although we are focused on multidimensional BSDEs, our results on uniqueness and stability are new even for one-dimensional BSDEs. As by-product, in the second part of the paper we establish the existence and uniqueness of Sobolev solutions to systems of (possibly) degenerate semilinear parabolic partial differential equations (PDEs). This is done with a nonlinear term of superlinear growth and a p-integrable terminal condition (p > 1). We cover certain systems of PDEs arising in physics and in particular the so-called logarithmic nonlinearity, u log(vertical bar u vertical bar). The proof we give for the uniqueness is new and rather nonstandard. Indeed, we introduce a new method which consists of deducing the uniqueness of the semilinear PDE from the uniqueness of its associated BSDE. In particular, we show by using BSDEs that uniqueness for a system of nonhomogeneous semilinear PDEs can be derived from the uniqueness for the homogeneous PDE satisfied by its associated linear part.