A note on backward stochastic differential equation with generator $f(y)|z|^2$

In this note, we consider the backward stochastic differential equation (BSDE) with generator $f(y)|z|^2,$ where the function $f$ is defined on an open interval $D$ and locally integral. The existence and uniqueness of solution of such BSDE is explored for bounded or unbounded terminal variables. A comparison theorem and a converse comparison theorem of such BSDEs are obtained.