Non-semimartingale solutions of reflected BSDEs and applications to Dynkin games

We introduce a new class of reflected backward stochastic differential equations with two c`adl`ag barriers, which need not satisfy any separation conditions. For that reason, in general, the solutions are not semimartingales. We prove existence, uniqueness and approximation results for solutions of equations defined on general filtered probability spaces. Applications to Dynkin games and variational inequalities, both stationary and evolutionary, are given.