1D nonlinear backward stochastic differential equations: a unified theory and applications

Since the celebrated paper by El Karoui, Peng and Quenez [Mathematical Finance, 7 (1997), 1--71], backward stochastic differential equations have found wide applications in stochastic control, financial technology and machine learning. In this paper, we present a comprehensive theory on the existence and uniqueness of adapted solutions to a one-dimensional nonlinear backward stochastic differential equation (1D BSDE for short), and assume that the generator $g$ has a unilateral linear or super-linear growth in the first unknown variable $y$, and has an at most quadratic growth in the second unknown variable $z$. We develop a unified methodology, featured by the test function method and the a priori estimate technique, to establish several existence theorems and comparison theorems, which immediately yield corresponding existence and uniqueness results. We also overview relevant known results and give some practical applications of our theoretical results. Finally, we list some open problems on the well-posedness of 1D BSDEs.