I have a question about the estimate of the exit probability on a finite interval. Given a $q$ function bounded and continuous, given the following SDE
\begin{cases}
dX_s=(\betaq(s))X_sds+\frac{1}{2}\beta^2(X_s)^2dW_s \\
X_t=y
\end{cases}
I should estimate the probability $\mathbb{P}\{\exists s \in [t,T] : (s,X_s) \in A \}$, where $A$ has this form $\{(t,x) \in [0,T] \times (0,+\infty) : 0 \leq y \leq L\}$.
Can someone help me with this estimate?
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$\begingroup$ Except when y=0, the answer is 1, because you have started it in that interval. I think the solution starting from 0 is always 0, and so it never enters the stricly positives. $\endgroup$– mikeApr 11, 2021 at 8:16

$\begingroup$ Thank you, even if in the article that I'm reading they state that the probability is less than c/y^2 where c is a positive constant, how can I justify it? $\endgroup$– RedLapmApr 11, 2021 at 10:05

$\begingroup$ I may not understand what A is. I thought A = $\{(t,x) \in [0,T] \times (0,+\infty) \} $ and the condition $ 0 \leq y \leq L$ didn't really belong in the braces, so that the probability you are looking for is the probability that the diffusion becomes positive before time T. $\endgroup$– mikeApr 12, 2021 at 7:12
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