Solution
= Solution
Apply the <Itô formula> to $f(t,x)=x_0e^{\sigma x+(\mu-\sigma^2/2)t}$ and the semimartingale vector $(t,B_t)$. Its derivatives give
$$
dX_t=\mu X_t\,dt+\sigma X_t\,dB_t.
$$
Therefore
$$
X_t=x_0\exp\left(\sigma B_t+
\left(\mu-\frac12\sigma^2\right)t\right).
$$
This is adapted to the given Brownian filtration and is consequently a strong solution; it is <geometric Brownian motion>.