Martingale-generated control structures and a framework for the dynamic programming principle
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This thesis constructs an abstract framework in which the dynamic programming principle (DPP) can be proven for a broad range of stochastic control problems. Using a distributional formulation of stochastic control, we prove the DPP for problems that optimize over sets of martingale measures. As an application, we use the classical martingale problem to prove the DPP for weak solutions of controlled diffusions, and use it show that the value function is a viscosity solution of the associated Hamilton-Jacobi-Bellman equation.