论文标题
平均场向后的随机微分方程和具有二次生长的非本地PDE
Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth
论文作者
论文摘要
在本文中，我们研究了具有二次生长的一般平均后向后随机微分方程（BSDES（BSDES）。 First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded.其次，通过Girsanov变换获得了一般平均BSDE的比较定理。第三，我们证明了粒子系统与二次生长的平均场BSDE的收敛性，并且还给出了收敛速率。最后，在此框架中，我们使用均值场BSDE为非局部偏微分方程（简称PDE）的粘度解提供了概率表示，作为扩展的非线性Feynman-kac公式，从而产生了对PDE的溶液的存在和独特性。
In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.
