A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order nonlinear variational inequalities with bilateral constraints. This type of inequalities arises in zero sum stochastic differential games of mixed type where each player uses both continuous control and stopping times. Under a nondegeneracy assumption Bensoussan and Friedman [1,4] have studied this type of problems. They proved the existence of a unique solution of these variational inequalities in certain weighted Sobolev spaces. This result together with certain techniques from stochastic calculus is then applied to show that the unique solution of these inequalities is the value function of certain stochastic differential games of mixed type. In this paper we study the same class of variational inequalities without the non-degeneracy assumption. The non-degeneracy assumption is crucially used in the analysis of the problem in [1,4]. Thus the method used in [1,4] does not apply to the degenerate case. We study the problem via the theory of viscosity solutions. We transform the variational inequalities with bilateral constraints to Hamilton–Jacobi–Isaacs (HJI for short) equations associated with a stochastic differential game problem with continuous control only. 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 variational inequalities. Then for a special case we identify this unique viscosity solution as the value function of the stochastic game with stopping times. We now describe our problem. Let Ui, i = 1,2, be the compact metric spaces. Let b : R ×U1 ×U2 → R d