Recent progress on systems of quadratic backward stochastic differential equations

A quadratic backward stochastic differential equation (BSDE) refers to an equation whose generator exhibits quadratic growth in the second unknown variable. In 1999, Peng listed the solvability of multidimensional quadratic BSDEs with unbounded terminal values" as an open problem, which remains partially solved and is still a challenging issue. This paper briefly introduces some recent progress in the theory and applications of multidimensional quadratic BSDEs over the past few years, focusing on several latest results within a non-Markovian framework where neither the terminal value nor the generator requires the assumptions of "sufficient smallness" or Malliavin-regularity. The main topics covered include: the existence and uniqueness of locally bounded solutions, globally bounded solutions, and globally unbounded solutions, as well as their applications in solving systems of backward stochastic Burgers equations and in studying the Nash equilibrium in a stochastic game.