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Let $W$ be a standard Brownian motion, and $a,b:\mathbb R_+\times \mathbb R\to\mathbb R$ be Lipschitz. Consider the stochastic differential equations:

$$X_t=1+\int_0^ta(s,X_s)ds + W_t,\quad\quad Y_t=1+\int_0^tb(s,Y_s)ds + W_t.$$

Fix some $T>0$ and set $\epsilon:=\|a-b\|:=\sup_{(t,x)\in [0,T]\times \mathbb R} |a(t,x)-b(t,x)|$. Denote by $\tau,\sigma$ the first hitting times of $X,Y$ at zero. What is the related literature for the estimation

$$\mathbb P[\tau\le t<\sigma]\le k(\epsilon),\quad \forall t\in [0,T]?$$

Here $k: \mathbb R_+ \to [0,1]$ is some continuous function vanishing at zero, e.g. $k(\epsilon)=C\epsilon^\alpha$ for some $C>0$ and $\alpha\in (0,1]$.

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  • $\begingroup$ An idea - use Gronwall as usual to bound the expected sup norm of $X - Y$ in terms of $\epsilon$, then Markov for a bound in probability, which translates to the desired bound on the probability in question. $\endgroup$
    – Nate River
    Commented May 7 at 11:04
  • $\begingroup$ @NateRiver Thanks for the comment. The "difference" between X and Y is straightforward, using classical arguments. However, the hitting time is not continuous with respect to the uniform norm. I don't see how the desired probability follows from the estimation of $X-Y$ $\endgroup$
    – GJC20
    Commented May 7 at 12:03
  • $\begingroup$ It is not continuous but it should be “continuous in probability”. The reasoning is that if $X$ hits zero at time $t$, with large probability it hits $-\varepsilon$ before time $t + \delta(\varepsilon)$, and so $Y$ hits zero sometime before $t + \delta(\varepsilon)$. An analysis of $\delta$ should give the form of $k$. $\endgroup$
    – Nate River
    Commented May 7 at 12:15
  • $\begingroup$ Ah of course, it seems $\delta$ can be chosen to be of order $\varepsilon^2$. $\endgroup$
    – Nate River
    Commented May 7 at 12:19
  • $\begingroup$ Shall we continue in private chat? I will see if I can explain it informally to you. If it is still not clear, then I can write up the details as an answer here. $\endgroup$
    – Nate River
    Commented May 7 at 12:23

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