Systems of quasilinear parabolic equations in ℝⁿ and systems of quadratic backward stochastic differential equations

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2]. One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on ℝⁿ , because of the lack of bounds. We give conditions so that the results of [2] can be extended to ℝⁿ . The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Z̆itković [8]. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in ℝⁿ  and not on a bounded domain. Xing and Z̆itković developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.