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 d Q d P 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 S t ≜ E Q [ ψ | 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.