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

Let Q and P be equivalent probability measures and let ψ be a J-dimensional vector of random variables such that dQdP and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale St≜EQ[ψ|Ft]. 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.