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