Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $ψ$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $ψ$ 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 $S_t \set \mathbb{E}^{\mathbb{Q}}[ψ|\mathcal{F}_t]$. While the drift $b=b(t,x)$ and the volatility $σ= σ(t,x)$ 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 $σ$ cannot be removed.

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