We provide a deterministic-control-based interpretation for a broad class of fullynonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in asmooth domain. We construct families of two-person games depending on a small parameterε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary conditionby introducing some specific rules near the boundary. We show that the value functionconverges, in the viscosity sense, to the solution of the PDE as ε tendsto zero. Moreover, our construction allows us to treat both the oblique and the mixed typeDirichlet–Neumann boundary conditions.
