Let $M$ be the space of right continuous functions $\ell: \mathbb R_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>t]$ for all $\ell \in M$ and $t\ge 0$, where $\tau^{\ell}:=\inf\{t\ge 0: X^{\ell}_t\le 0\}$ and

$$X^{\ell}_t:=1+t+\int_0^t\frac{1}{1+\ell(s)}dW_s,\quad \forall t\ge 0.$$

Here $(W_t)_{t\ge 0}$ denotes a Brownian motion. Let $M$ be endowed with the topology as follow: $\ell^n$ converges to $\ell$ in $M$ iff $\lim_{n\to\infty}\ell^n(t)=\ell(t)$ for all points of continuity of $\ell$. Can we prove the continuity of $\Gamma$ with respect to this topology?

Let $M$ be the space of right continuous functions $\ell: \mathbb R_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>t]$ for all $\ell \in M$ and $t\ge 0$, where $\tau^{\ell}:=\inf\{t\ge 0: X^{\ell}_t\le 0\}$ and

$$X^{\ell}_t:=1+t+\int_0^t\frac{1}{1+\ell(s)}dW_s,\quad \forall t\ge 0.$$

Here $(W_t)_{t\ge 0}$ denotes a Brownian motion. Let $M$ be endowed with the topology as follow: $\ell^n$ converges to $\ell$ in $M$ iff $\lim_{n\to\infty}\ell^n(t)=\ell(t)$ for all points of continuity of $\ell$. Can we prove the continuity of $\Gamma$ with respect to this topology?

Remark : To prove $\{\inf_{0\le s\le t}X^{\ell}_s\le 0\}=\{\inf_{0\le s\le t}X^{\ell}_s< 0\}$, it suffices to use Lévy's characterization. More precisely, we can write $X^{\ell}_t=1+t+B_{\langle X^{\ell}\rangle_t}\equiv 1+t+B_{L(t)}$, where $B$ denotes a Brownian motion and $L(t):=\int_0^t ds/(1+\ell(s))^2$. Therefore

$$\inf_{0\le s\le t}X^{\ell}_t = \inf_{0\le u\le L(t)}\{1+L^{-1}(u)+B_t\},$$

which implies the desired result as $\inf_{0\le u\le L(t)}\{1+L^{-1}(u)+B_t\}$ admits a density.