We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators that are singular in y. First, we establish the existence of solutions and a comparison theorem, thereby extending the existing results in the literature. Furthermore, we analyze the stability properties, derive the Feynman–Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semi-linear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and the certainty equivalent based on g-expectation. In these applications, the quadratic coefficients in the generators, respectively, quantify ambiguity aversion and absolute risk aversion.
Wang et al. (Thu,) studied this question.