On the backward stochastic differential equation with generator $f(y)\mathrm{\mid }z{\mathrm{\mid }}^2$

In this paper, we consider the backward stochastic differential equation (BSDE) with generator $f(y)\mathrm{\mid }z{\mathrm{\mid }}^2,$ where the function $f$ is defined on an open interval $D$ and locally integrable. The existence and uniqueness of bounded solutions and $L^p(p\ge 1)$ solutions of such BSDEs are obtained. Some comparison theorems and a converse comparison theorem of such BSDEs are established. As an application, we give a probabilistic interpretation of viscosity solution of quadratic PDEs.