非线性形式的倒向随机微分方程（简写为BSDE）由我国彭实戈院士及其合作者于1990年提出并迅速成为国际研究热点，在随机控制和偏微分方程（简写为PDE）等领域具有重要应用。BSDE理论的一个研究热点是非线性Feynman-Kac公式，即通过BSDE理论研究非线性PDE粘性解的概率解释及性质。目前，由于非线性Neumann边值问题相对复杂，其粘性解概率解释的相关研究相对较少。本项目旨在非Lipschitz条件下借助递归形式的状态受限随机微分对策问题研究Isaacs方程非线性Neumann边值问题的粘性解存在性和唯一性，从而建立Isaacs方程非线性Neumann边值问题的Feynman-Kac公式。本项目拟借助BSDE生成元表示定理方法证明此粘性解的概率解释，揭示Isaacs方程粘性解的概率解释问题与BSDE生成元表示问题之间的关系，为PDE粘性解存在性理论提供新的视角和思考方向。

The nonlinear Backward Stochastic Differential Equations (BSDEs for abbreviated) were proposed by academician Shige Peng and his coworkers in 1990. This theory became a hotspot of research promptly because its applications to various fields, such as stochastic optimal control and partial differential equation (PDE for abbreviated). A remarkable application of BSDE theory is nonlinear Feynman-Kac formula, which can be interpreted as studying probabilistic interpretation in viscosity sense for solutions of PDEs, or related properties, by BSDE theory. Currently, the research of probabilistic interpretation for nonlinear Neumann boundary value problems in the viscosity sense is insufficient because nonlinear Neumann problems are more complicated. The major object of this project is to study the existence and uniqueness for viscosity solutions of Isaacs equations with nonlinear Neumann boundary value problems under non-Lipschitz conditions utilizing stochastic differential game with state constraints and recursive cost functionals, thereby obtaining the Feynman-Kac formula for Isaacs equations with nonlinear Neumann problems. This project will prove the probabilistic interpretation for viscosity solutions using the representation theorem for generators of BSDEs. This approach can not only clarify the essential relationship between the probabilistic interpretation for solutions of Isaacs equations in viscosity sense and the representation theorem for generators of BSDEs, but also provides a novel perspective for the existence of viscosity solutions of PDEs.