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In this paper, we consider the backward stochastic differential equation(BSDE) with generator $f(y)|z|^2,$ where the function $f$ is defined on an openinterval $D$ and locally integrable. The existence and uniqueness of boundedsolutions and $L^p(p\geq1)$ solutions of such BSDEs are obtained. Somecomparison theorems and a converse comparison theorem of such BSDEs areestablished. As an application, we give a probabilistic interpretation ofviscosity solution of quadratic PDEs.

On the backward stochastic differential equation with generator $f(y)|z|^2$