The theory of backward stochastic differential equation (BSDE) is one of the main tools to study continuous stochastic optimization. In this talk, we will review
1) the relationship between BSDE and stochastic optimization/stochastic game/PDE,
2) the well-posedness results of BSDEs, and
3) recent developments in a system of quadratic BSDEs.
In particular, a new existence and uniqueness result for a system of non-Markovian quadratic BSDEs will be presented. Such BSDE appears in stochastic differential game (Carmona, 2016) and price impact model (Kramkov and Pulido, 2016). Moreover, the result generalizes the well-posedness of quasilinear PDE of Ladyzhenskaia and Uraltseva (1968) to that of the corresponding path-dependent PDE.