A strong comparison result for the bellman equation arising in stochastic exit time control problems and its applications

this article, we consider the Dirichlet problem for the fully nonlinear, possibly degenerate, Hamilton-Jacobi-Bellman equation arising in stochastic optimal control with exit time. Our main contribution is to provide for this problem a rather general "Strong Comparison Result" i.e. a maximum principle type result for discontinuous viscosity solutions. The motivation for proving such a result is that it is a key argument when one wants to establish that