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Let $C(u,s)$ be cost functional for an admissible control $u$ with initial state of the system being $s$.
Our aim is the solution of the following problem: $$\inf_u E(C(u,s))$$

We defined the value $V$ function, which equals to expected remaining costs of optimally controlled on $(t,T]$

$$V(t,s)=\inf_u E\left[\int_t^T{C(u_r,s)}\,dr\right]$$

Then we can define a process: $$M_t=C_t(u,s)+V(t, s)$$

By martingale optimality principle, for any admissible startegy $u$, $M_t$ is a submartingale and for optimal $u^*$ it is a martingale. From this condition we can find an optimal strategy.

Now in verification argument we guess the $V$ function, but why this really gives us the optimal strategy? In other words, when we check that the strategy is truely optimal without using the guess?

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I changed \underset{u}\text{inf} to \inf_u. In a "displayed" setting that has the same effect except that it ALSO automatically provides proper spacing before and after $\inf$ in expressions like $\displaystyle a\inf_{u\in\mathbb R} b$. (In an "inline" setting, the subscript appears below and to the right of $\inf$ unless \displaystyle is used.) Standard operator names like \inf generally have all the typesetting conventions built-in to them that are conventional for the symbol in question. – Michael Hardy Apr 13 '13 at 16:57
@MichaelHardy Thanks a lot. I will remember this. – type2 Apr 13 '13 at 17:37

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