Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ with $\|\alpha\|_T:=\max_{0\le t\le T}|\alpha(t)|+\max_{0\le t\le T}|\alpha'(t)|\le 1$. For any $\alpha\in C^1([0,T])$, consider
$$dX_{\alpha,t} = b\big(t,X_{\alpha,t},\alpha(t)\big)dt + W_t,\quad \forall t\ge 0\quad\quad\quad\quad\quad\quad (\ast)$$
and denote $(X^{t,x}_{\alpha,s})_{s\ge t}$ the solution to $(\ast)$ with initial condition $X^{t,x}_{\alpha,t}=x$. Define the operator $\Gamma: \alpha\to\Gamma[\alpha]$ by
$$\Gamma[\alpha](s):=\mathbb P\big[X^{0,x}_{\alpha,s}>0\big] + \int_0^s \mathbb P\big[X^{t,0}_{\alpha,s}>0\big]\alpha'(t)dt,\quad \forall s\in [0,T].$$
Can we find some continuous function $\theta:\mathbb R_+\to\mathbb R_+$ with $\theta(0)=0$ s.t.
$$\|\Gamma[\alpha]-\Gamma[\beta]\|_T \le \theta(T)\|\alpha-\beta\|_T?$$
Any answer, comments and references are highly appreciated.
PS : The difficulty for me is to estimate the Lipschitz continuity of $\partial_s \mathbb P\big[X^{0,x}_{\alpha,s}>0\big]$ and $\partial_s \mathbb P\big[X^{t,0}_{\alpha,s}>0\big]$ on $\alpha$. It seems that Girsanov's theorem may be helpful...
