Solution
= Solution
In a <Black-Scholes model>, the risk-free asset and risky asset satisfy
$$
dS_t^0=rS_t^0\,dt,
\qquad
dS_t=\mu S_t\,dt+\sigma S_t\,dB_t,
$$
where $B$ is a <Brownian motion>, $\sigma>0$, and the coefficients are constant. With $S_0^0=1$,
$$
S_t^0=e^{rt},
\qquad
S_t=S_0\exp\left((\mu-\tfrac12\sigma^2)t+\sigma B_t\right),
$$
so the risky asset is a <geometric Brownian motion>.