Stochastic control problem, admissible control, value function क्या है?

A stochastic control problem chooses a process $u_t$ that influences the dynamics of a state $X$, here $dX_t = \mu(t, X_t, u_t)\,dt + \sigma(t, X_t, u_t)\,dW_t$ or its discrete-time analogue, to maximise $J(t, x; u) = \E_{t,x}[\int_t^Tf(s, X_s, u_s)\,ds + g(X_T)]$. An admissible control is an adapted process with values in a set $\mathcal U$ for which the state equation has a solution and $J$ is well defined. The value function is $V(t, x) = \sup_uJ(t, x; u)$ over admissible controls.