= Solution
On a <filtered probability space> carrying a standard <Brownian motion> $W$ with independent increments relative to the <filtration>, a strictly positive security price follows <geometric Brownian motion> with constant drift $\mu$ and volatility $\sigma$ if it solves the <stochastic differential equation>
$$
\boxed{dS_t=\mu S_t\,dt+\sigma S_t\,dW_t,\qquad S_0>0.}
$$
Here $W_t-W_s$ is independent of $\mathcal F_s$ and has the <normal distribution> $N(0,t-s)$ for $s<t$. The solution is
$$
S_t=S_0\exp\{(\mu-\sigma^2/2)t+\sigma W_t\}.
$$
Indeed applying the <Itô formula> to this exponential gives the specified drift and volatility. Consequently the price is positive and continuous, and
$$
\log(S_t/S_s)\mid\mathcal F_s\sim
N((\mu-\sigma^2/2)(t-s),\sigma^2(t-s)).
$$
Thus disjoint log-return increments are independent and stationary. The parameter $\mu$ is the instantaneous expected relative return under the physical <probability measure>, not the mean log-return. Under the pricing <risk-neutral measure> it becomes the risk-free rate for a non-dividend-paying <stock>.
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