= Conditional mean of a log-mean-reverting price
{title2=$F(t,T)=\exp[a\log S_t+(1-a)\widetilde\mu+\sigma^2(1-a^2)/(4\alpha)]$}
For a <log-mean-reverting price>, set $a=e^{-\alpha(T-t)}$. The <Ornstein-Uhlenbeck transition distribution> gives conditional log-mean $a\log S_t+(1-a)\widetilde\mu$ 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> yields
$$
dF(t,T)=\sigma e^{-\alpha(T-t)}F(t,T)dW_t.
$$
This 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.
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