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 ${\mathbb{R}}^n$, because of the lack of bounds. We give conditions so that the results of [2] can be extended to ${\mathbb{R}}^n$. 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 ${\mathbb{R}}^n$ 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.

本文的目的是双重的。第一个目标是完成Bensoussan和Frehse的先前工作[2]。本文的一个主要限制是它们是PDE系统。在有界域上。人们可以期待解决方案的局限性，因为人们寻求的是平滑的解决方案。这对于方法的开发是非常重要的性质。确实，存在于有界域中的解决方案可能无法存在于${\mathbb{[R}}^{ñ}$，因为缺少界限。我们给出条件，以便可以将[2]的结果扩展到${\mathbb{[R}}^{ñ}$。第二个目标是考虑PDE系统的BSDE（反向随机微分方程）版本。这是Xing和Z̆itković[8]近期工作的目标。他们考虑了具有二次增长的BSDE系统，这是BSDE文献中一个众所周知的开放问题。由于BSDE是马尔可夫式的，因此问题等同于解析问题。但是，由于这种动机，分析问题在${\mathbb{[R}}^{ñ}$而不是在有界域上。Xing和Zitkovkov提出了一种概率方法。分析问题和BSDE之间的联系并不明显。我们的目标是证明分析方法可以完全转换为一种概率方法。然而，概率概念在转换为分析框架后也很有用。对于唯一性结果尤其如此。