My formula is
\begin{align} \mathbb{P}\big(t-\Delta t~\leq\text{log} \int_{0}^{t} \sigma_s^2 ~ds \leq t+\Delta t\big) & = \int_{t-\Delta t}^{t+\Delta t} f_{\text{log} \int_{0}^{t} \sigma_s^2 ~ds}~ (y) ~dy\\ & = \int_{t-\Delta t}^{t+\Delta t} f_{G}~ (y) ~dy\\ & = \int_{-\infty}^{t+\Delta t} f_{G}~ (y) ~dy-\int_{-\infty}^{t-\Delta t} f_{G}~ (y) ~dy\\ &=\int_{x}^{t+\Delta t} f_{G}~ (y) ~dy-\int_{x}^{t-\Delta t} f_{G}~ (y) ~dy \end{align}
I am going to compute the probability $\mathbb{P}(\cdot)$. Therefore, $\int_{x}^{t+\Delta t} f_{G}~ (y) ~dy$ and $\int_{x}^{t-\Delta t} f_{G}~ (y) ~dy$ can be greater than 1? Is it correct?
Thank you!