This paper studies the fully nonlinear first order partial differential equation \(u+H(x,Du)=0\) in a bounded domain \(\theta\) with \(u=g\) on the boundary of \(\theta\) using the connection between viscosity solutions and differential games or control theory. Here the control problem involves the exit time from the domain. The main result is the existence of a unique viscosity solution of the equation if there are continuous viscosity sub and super solutions in \(\theta\). The method of proof is to use a cutoff function and then apply the known viscosity theorems in \(R^ n.\) In the second part of this paper an infinitesimal dynamic programming principle is derived which is applied later to find piecewise linear approximate characteristics along with the value function is nonincreasing. This results in another, more restricted derivation of the main result mentioned above.