Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space on which a Brownian motion $W$ is defined. For each $n\ge 1$, let $U_n:=[1/n,n]$ and $\mathcal U_n$ be the set of progressively measurable (w.r.t. the Brownian filtration) processes $p=(p_t)_{t\ge 0}$ taking values in $U_n$. Consider the following stochastic control problem: for $t\in [0,1]$ and $x\in [0,1]$,

$$v_n(t,x):=\sup_{p\in \mathcal U_n}\mathbb E\left[\int_t^{\min(1,\tau^{p,t,x})}\big(1+\log(p_s)\big)du\right],$$

where $dX^{p,t,x}_s=\sqrt{2p_s}dW_s$ for all $s\ge t$ with $X^{p,t,x}_t=x$ and $\tau^{p,t,x}:=\{s\ge t: X^{p,t,x}_s\notin (0,1)\}$.

Set $U_\infty:=\mathbb R_+$, and define similarly $\mathcal U_\infty$ and the corresponding stochastic control problem $v_\infty(t,x)$. I have two questions :

1. Does the pointwise convergence $v_n\to v_\infty$ hold?
2. Does there exist $N$ large enough s.t. $v_n=v_N$ for all $n\ge N$?