Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space on which a Brownian motion $W$ is defined, and $\mathcal U$ be the set of progressively measurable (w.r.t. the Brownian filtration) processes $p=(p_t)_{t\ge 0}$ taking values in $\mathbb R_+$ s.t.
$$\mathbb E\left[\int_0^T p_t dt\right]<\infty,\quad \forall T\ge 0. $$
For each $n\ge 1$, denote by $\mathcal U_n\subset \mathcal U$ the subset of $p=(p_t)_{t\ge 0}$ taking values in $[1/n,\infty)$. Define, for $p\in\mathcal U$, $t\in [0,1]$ and $x\in [0,1]$,
\begin{eqnarray} J(t,x,p):=\mathbb E\left[\int_t^{\min(1,\tau^{p,t,x})}\big(1+\log(p_s)\big)ds\right] \end{eqnarray}
where $\tau^{p,t,x}:=\{s\ge t: X^{p,t,x}_s\notin (0,1)\}$ and $dX^{p,t,x}_s=\sqrt{2p_s}dW_s$ for all $s\ge t$ with $X^{p,t,x}_t:=x$. Can we prove the existence of some $N$ s.t. for any $p\in\mathcal U$, there exists $q\in\mathcal U_N$ satisfying $J(t,x,p)\le J(t,x,q)$?
