  The present paper is devoted to the study of diagonally quadratic backward
stochastic differential equation with oblique reflection. Using a penalization
approach, we show the existence fo a solution by providing some delicated a
priori estimates. We further obtain the uniqueness by verifying the first
component of the solution is indeed the value of a switching probelm for
quadratic BSDEs. Moreover, we provide an extension for the solvability and
apply our results to study a risk-sensitive switching problem for functional
stochastic differential equations.
