= Solution
For $x_t=\log S_t$, the <Itô formula> uses $(\log S)'=1/S$ and $(\log S)''=-1/S^2$. Consequently
$$
dx_t=\frac{dS_t}{S_t}-\frac12\frac{d\langle S\rangle_t}{S_t^2}=\left[\alpha(\mu-x_t)-\frac12\sigma^2\right]dt+\sigma\,dZ_t.
$$
The <quadratic variation> term shifts the mean level. Thus the <log-mean-reverting price> has logarithm following the <Ornstein-Uhlenbeck process>
$$
\boxed{dx_t=\alpha(\widetilde\mu-x_t)dt+\sigma\,dZ_t,\qquad \widetilde\mu=\mu-\frac{\sigma^2}{2\alpha}}.
$$
It is the logarithm, rather than the price itself, which reverts linearly to a constant mean level.
Back to article page
