  We consider a class of backward stochastic differential equations (BSDEs)
driven by Brownian motion and Poisson random measure, and subject to
constraints on the jump component. We prove the existence and uniqueness of the
minimal solution for the BSDEs by using a penalization approach. Moreover, we
show that under mild conditions the minimal solutions to these constrained
BSDEs can be characterized as the unique viscosity solution of
quasi-variational inequalities (QVIs), which leads to a probabilistic
representation for solutions to QVIs. Such a representation in particular gives
a new stochastic formula for value functions of a class of impulse control
problems. As a direct consequence, this suggests a numerical scheme for the
solution of such QVIs via the simulation of the penalized BSDEs.
