= Log-mean-reverting price
{title2=$dS_t=\alpha(\mu-\log S_t)S_tdt+\sigma S_tdW_t$}
For $\alpha>0$, $\sigma>0$, this positive <stochastic process> is the exponential of an <Ornstein-Uhlenbeck process>. The <Itô formula> gives
$$
d\log S_t=\alpha(\widetilde\mu-\log S_t)dt+\sigma dW_t,\qquad\widetilde\mu=\mu-\frac{\sigma^2}{2\alpha}.
$$
The mean level of the logarithm differs from $\mu$ by the <quadratic variation> correction. In a <stock> model, specifying this under a physical probability measure does not by itself specify <risk-neutral pricing>. A non-dividend-paying traded <stock> has drift $rS$ under a <risk-neutral measure>; a different drift under that measure would require a corresponding dividend or carry convention.
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