  In this paper, we introduce a non-linear Snell envelope which at each time
represents the maximal value that can be achieved by stopping a BSDE with
constrained jumps. We establish the existence of the Snell envelope by
employing a penalization technique and the primary challenge we encounter is
demonstrating the regularity of the limit for the scheme. Additionally, we
relate the Snell envelope to a finite horizon, zero-sum stochastic differential
game, where one player controls a path-dependent stochastic system by invoking
impulses, while the opponent is given the opportunity to stop the game
prematurely. Importantly, by developing new techniques within the realm of
control randomization, we demonstrate that the value of the game exists and is
precisely characterized by our non-linear Snell envelope.
