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### Is HJB equation a local or global optimum?

Posted: January 29th, 2015, 5:27 am
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.

### Is HJB equation a local or global optimum?

Posted: March 5th, 2015, 11:57 am
The funny thing here is that you scratch your head only as you forgot the assumptions. You are completely right, a sequence of local optima doesn't make up to the global optimum. If it were like this optimization theory would be complete at the level of greedy functions.However, the HJB (or its discrete version, the Bellman equation) is exactly designed for those kind of problems, where it is exactly like this, i.e. where a sequence of local optima leads to the global optimum. The same in another terms: the property an optimization problem needs to satisfy in order to be treated by the dynamic programming method is "Optimal substructure".