I am a bit confused on the HJB equation. In the derivation of Hamilton?Jacobi?Bellman (HJB) equation, we take the first order optimality at each time step, so that the trading strategy is optimum over the next infinitesimal time step. It seems that this strategy gives only local optimum, which is not necessarily a global optimum.However, from the verification theorem (www.math.nyu.edu/faculty/kohn/pde.finan ... ction4.pdf
), it seems to be a global optimum. On the other hand, from WIKI ?When solved locally, the HJB is a necessary condition, but when solved over the whole of state space, the HJB equation is a necessary and sufficient condition for an optimum.? So what?s the difference between solving locally and globally?Thank you.