 Log-mean-reverting price

For $\alpha >0$, $\sigma >0$, this positive stochastic process is the exponential of an Ornstein-Uhlenbeck process. The Itô formula givesThe 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.