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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / d / Solution

{title2=$dX_t=\mu X_t\,dt+\sigma X_t\,dB_t$}
{wiki}

Geometric Brownian motion has the explicit strong solution
$$
X_t=X_0\exp\left(\sigma B_t+
\left(\mu-\frac12\sigma^2\right)t\right).
$$


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 Incoming links: Geometric Brownian motion