Consider the following stochastic integral: $$X_{t}^{(\epsilon)}=\epsilon e^{t}\int_{0}^{t}e^{-s}dW_{s},\ \ \ X_{0}=0.$$ Define the hitting time $\tau_{\epsilon}:=\inf\{t\geq 0:|X_{t}^{(\epsilon)}|=1\}$. What is the distribution of $\tau_{\epsilon}$?
2026-03-29 08:35:24.1774773324
Limiting distribution of a first passage time.
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