= Solution
A <futures contract> fixes a maturity and underlying quantity, but its quoted delivery price is marked to market: gains and losses from quote changes are settled through the margin account, typically daily. A newly settled position has zero contract value; the quoted futures price is not an upfront purchase price for that position. At maturity the quote equals the spot price. In the continuous-settlement idealization, discounted futures gains must be local <martingales> under the money-market <risk-neutral measure>. Thus the quote has zero pricing-measure drift. Assuming the relevant <integrability> makes it a true <martingale>, \b[$\boxed{F_{tT}=\mathbb E_Q[S_T\mid\mathcal F_t]}$.]
This is a <futures pricing> formula. In contrast, an unsettled <forward contract> has delivery quote $\mathbb E_Q[D_{tT}S_T\mid\mathcal F_t]/\mathbb E_Q[D_{tT}\mid\mathcal F_t]$, where $D_{tT}=e^{-\int_t^T r_udu}$. Stochastic <interest rates> can make the two quotes differ. <Contango> means the futures quote is above spot for the maturity considered, while <backwardation> means it is below spot; futures curves are correspondingly described as rising or falling when comparing maturities.
For the <Multivariate Ornstein-Uhlenbeck process>, multiplication by the <matrix exponential> $e^{At}$ gives $d(e^{At}X_t)=e^{At}dB_t$, hence
$$
\boxed{X_t=e^{-At}X_0+\int_0^t e^{-A(t-u)}dB_u.}
$$
For deterministic $X_0=x_0$, the <Gaussian distribution> has mean $e^{-At}x_0$ and <covariance>
$$
C_t=\int_0^t e^{-Au}e^{-A^{\mathsf T}u}\,du.
$$
Equivalently $\dot C=I-AC-CA^{\mathsf T}$, $C_0=0$. No symmetry or invertibility of $A$ is required. This <covariance> can also be evaluated from a single block <matrix exponential>: if the upper-right block of $\exp(t\left(\begin{smallmatrix}-A&I\\0&A^{\mathsf T}\end{smallmatrix}\right))$ is $H_t$, then $C_t=H_te^{-A^{\mathsf T}t}$. If $X_0$ is random, these assertions hold conditionally on $X_0$; the unconditional law need not be Gaussian. A stationary Gaussian law exists when the <eigenvalues> of $A$ have positive real parts, but stability is unnecessary for the finite-time formulas.
For $\tau=T-t$, the conditional mean of $X_T$ is $e^{-A\tau}X_t$ and its conditional <covariance> is $C_\tau$. The Gaussian exponential-moment formula therefore gives \b[the explicit quote]
$$
\boxed{F_{tT}=\exp\left(b^{\mathsf T}e^{-A\tau}X_t+\tfrac12 b^{\mathsf T}C_\tau b\right).}
$$
It tends to $e^{b\cdot X_t}=S_t$ as $T\downarrow t$, as required.
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