  Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let
$\psi$ be a $J$-dimensional vector of random variables such that
$\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $\psi$ are defined in terms of a weak
solution $X$ to a $d$-dimensional stochastic differential equation. Motivated
by the problem of \emph{endogenous completeness} in financial economics we
present conditions which guarantee that every local martingale under
$\mathbb{Q}$ is a stochastic integral with respect to the $J$-dimensional
martingale $S_t \set \mathbb{E}^{\mathbb{Q}}[\psi|\mathcal{F}_t]$. While the
drift $b=b(t,x)$ and the volatility $\sigma = \sigma(t,x)$ coefficients for $X$
need to have only minimal regularity properties with respect to $x$, they are
assumed to be analytic functions with respect to $t$. We provide a
counter-example showing that this $t$-analyticity assumption for $\sigma$
cannot be removed.
