Consider a probability space $(\Omega, \mathcal F , \mathbb P)$, $(B_t)_{t\geq0}$ M-dimentional brownian motion adapted to a filtration $(\mathcal F_t)_{t\geq0}$ over $\Omega$. In this context consider a controlled SDE
$$ X_s = x + \int _t ^s b(X_u, \alpha _u) ~ds + \int _t ^s \sigma(X_u, \alpha _u) ~dBs $$
where $\alpha \in \mathcal A(t) = \{ \alpha : [t, +\infty]\times \Omega \rightarrow A\ ; \alpha \ \text{progressively measurable to }(\mathcal F_s)_{s>t} \}$, $A$ is a compact subspace of finite dimension. Also, $b:\mathbb R ^d \times A \rightarrow \mathbb R ^d$ and $\sigma :\mathbb R ^d \times A \rightarrow \mathbb R ^{d\times M}$ are bounded and globally Lipschitz relative to the first variable uniformly relative to the second one, wich garantie the existence of a unique solution noted $X_s^{t,x,\alpha}$.
Now consider an also globally Lipschitz function $g : \mathbb R ^d \rightarrow \mathbb R $ such that $1\leq g(x)\leq 2, \ \forall x \in \mathbb R ^d$
I'm interested in solving the following problem $$ \overline{V}(t,x) := \inf _{\alpha \in \mathcal A (t)} \text{ess-sup}_\Omega ~ g (X_T^{t,x,\alpha})$$ for a fixed $T>0$.
Since the characterization of $\overline V $ is delicate, I desire to approach it by the following sequence of functions
$$ V_n(t,x) := \inf _{\alpha \in \mathcal A (t)} (\mathbb E [( g(X_T^{t,x,\alpha}))^n])^{1/n}, \quad \forall n \in \mathbb N^*.$$
We can show that fora all $n \leq m $ in $\mathbb N ^*$ we have $$ V_n(t,x) \leq V_m(t,x) \leq \overline V(t,x)$$
The I desire to show that the limit of $V_n(t,x)$ exists and that verifies
$$\lim_{n \rightarrow + \infty} V_n(t,x) = \inf \{ a \in \mathbb R : \forall \varepsilon >0, \exists \alpha \in \mathcal A(t), \exists E \in \mathcal F \text{ with } \mathbb P [E] \geq 1- \varepsilon \text{ and } \\g(X_T^{t,x,\alpha}) \leq a + \varepsilon, \mathbb P \text{-a.e. on } E \}$$
All advices are appreciated. Thanks in advance.