In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in continuous time problem, to derive sufficient condition under which DPP is valid? Is there an easy way to explain why measurable selection theorem is important in DPP?

For ex., here is one control problem: Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F})$ be a filtered probability space. Let $\mathcal{A}$ be admissible control space, and $X^{t,x,\alpha}(s)$ be a controlled Markov process with initial $(t,x)$ and control $\alpha \in \mathcal{A}$. The value function is $V(t,x) = \inf \mathbb{E}[ g(X^{t,x, \alpha}(\tau)) ]$, where $\inf$ is over $\alpha \in \mathcal{A}$, and $\tau$ is a given stopping time. The question is, what is the sufficient condition for $V(\cdot)$ to have $V(t,x) = \inf \mathbb{E}[V(\theta, X^{t,x,\alpha}(\theta))]$ for all stopping time $\theta \in [t,\tau]$?


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