HOMOGENIZATION OF BSDES WITH TWO REFLECTING BARRIERS, VARIATIONAL INEQUALITY AND STOCHASTIC GAME

In this paper, we study the limit of semilinear variational inequality with bilateral constraints and stochastic differential games of mixed type. By a penalization method, we first study homogenization properties for system of two barriers reflected backward stochastic differential equation in the Markovian setting. This result together with certain techniques from stochastic calculus is then applied to show that the unique solution of the homogenized problem is also the value function of certain stochastic differential games of mixed type. Then using standard results from the theory of viscosity solutions, we show that the value function of this stochastic differential game with continuous control is the unique viscosity solution of the corresponding limit semilinear variational inequalities too.