# Exit probability on a finite interval

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
$$\begin{cases} dX_s=(\beta-q(s))X_sds+\frac{1}{2}\beta^2(X_s)^2dW_s \\ X_t=y \end{cases}$$ 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?

• 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.
– mike
Apr 11, 2021 at 8:16
• 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? Apr 11, 2021 at 10:05
• 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.
– mike
Apr 12, 2021 at 7:12