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 $(t,T]$
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?