| The aim of this paper is to provide an original step-by-step proof of the optimality principle, for a certain class of stochastic control problems with exit time. The presence of exit time is responsible of measurability problems involving the control processes. Therefore, in order to prove the dynamic programming principle, we proceed by measurable selection. The measurable selection theorem comes from an important result in functional analysis due to Jankov and von Neumann, and it is also obtained by formalizing the main properties of the admissible region and of the state variable. |
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