 Conditional mean of a log-mean-reverting price

For a log-mean-reverting price, set $a=e^{-\alpha (T-t)}$. The Ornstein-Uhlenbeck transition distribution gives conditional log-mean $a\log S_t+(1-a)\tilde{\mu }$ and log-variance ${\sigma }^2(1-a^2)/(2\alpha )$. The moment-generating function of a normal distribution therefore gives the displayed formula for $F(t,T)=\mathbb{E}[S_T\mid {\mathcal{F}}_t]$. It is a martingale in $t$ for fixed $T$, by the tower property of conditional expectation, and the Itô formula yieldsThis uses the same probability measure as the specified conditional expectation. It describes futures pricing only when that measure is an appropriate pricing measure, with the required cash-flow conventions. For a non-dividend-paying traded stock and constant interest rate, the forward contract delivery price follows the usual carry relation instead.