$dX_t = \mu X_t dt + \sigma X_t dB_t$ with $X_0 = x_0 > 0$. Apply Itô to $f(X_t) = \log X_t$:

$d(\log X_t) = \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma dB_t.$

Integrating:

$X_t = x_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma B_t\right).$

This is the Black-Scholes model for stock prices.

$dX_t = -\theta(X_t - \mu) dt + \sigma dB_t$. This models a variable reverting to mean $\mu$ with rate $\theta$. Solution:

$X_t = \mu + (X_0 - \mu)e^{-\theta t} + \sigma \int_0^t e^{-\theta(t-s)} dB_s.$

As $t \to \infty$, $X_t$ converges in distribution to $\mathcal{N}(\mu, \sigma^2/(2\theta))$.