Probability of a Brownian Motion to fall in a bandwidth

Question

Let $X_t$ be defined as

$$ X_t = X_0+\int_0^t\sigma_{0}\,dW_s, $$

where $W_s$ is a Wiener process and $\sigma_0\in\mathbb{R}^{+}/{0}$. Which is the probability

$$ \mathbb{P}\left[a<X_t-X_0<b\right] ? $$

Answer 1

$\int_0^1 \sigma_0 dW_s = \sigma_0 W_t$ is a centered normal variable with variance $\sigma_0^2t$