Constrained BSDEs, viscosity solutions of variational inequalities and their applications

In this paper, we study the relation between the smallest $g$-supersolution of constrained backward stochastic differential equation and viscosity solution of constraint semilinear parabolic PDE, i.e. variation inequalities. And we get an existence result of variation inequalities via constrained BSDE, and prove a uniqueness result with a condition on the constraint. Then we use these results to give a probabilistic interpretation result for reflected BSDE with a discontinuous barrier and other kind of reflected BSDE.